Linear Equations in two variables

The correct answer is \(4.51\). It’s given that the equation \(4.51 x + 6.07 y = 896.86\) represents this situation, where \(x\) is the number of smaller containers sold, \(y\) is the number of larger containers sold, and \(896.86\) is the store’s total sales, in dollars, of blueberries last month. Therefore, \(4.51 x\) represents the store's sales, in dollars, of smaller containers, and \(6.07 y\) represents the store's sales, in dollars, of larger containers. Since \(x\) is the number of smaller containers sold, the price, in dollars, of each smaller container is \(4.51\).

Choice D is correct. The linear relationship between \(x\) and \(y\) can be represented by the equation \(y = m x + b\), where \(m\) is the slope of the graph of this equation in the xy-plane and \(b\) is the y-coordinate of the y-intercept. The slope of a line between any two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line can be calculated using the slope formula \(m = y_2 -y_1/x_2 -x_1\). Based on the table, the graph contains the points \((-18, -48)\) and \((7, 52)\). Substituting \((-18, -48)\) and \((7, 52)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the slope formula yields \(m = 52 -(-48)/7 -(-18)\), which is equivalent to \(m = 100/25\), or \(m = 4\). Substituting \(4\) for \(m\), \(-18\) for \(x\), and \(-48\) for \(y\) in the equation \(y = m x + b\) yields \(-48 = 4(-18)+ b\), or \(-48 = -72 + b\). Adding \(72\) to both sides of this equation yields \(24 = b\). Therefore, \(m = 4\) and \(b = 24\). Substituting \(4\) for \(m\) and \(24\) for \(b\) in the equation \(y = m x + b\) yields \(y = 4 x + 24\). Thus, the equation \(y = 4 x + 24\) represents the linear relationship between \(x\) and \(y\). It's also given that the graph of the linear equation representing this relationship in the xy-plane passes through the point \((1 seventh, a)\). Substituting \(1 seventh\) for \(x\) and \(a\) for \(y\) in the equation \(y = 4 x + 24\) yields \(a = 4(1 seventh)+ 24\), which is equivalent to \(a = 4 sevenths + 168/7\), or \(a = 172/7\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. Let \(p\) represent the number of pounds of potatoes and let \(c\) represent the number of pounds of celery that Davio bought. It’s given that potatoes cost \($ 0.69\) per pound and celery costs \($ 0.99\) per pound. If Davio spent \($ 5.34\) in total, then the equation \(0.69 p + 0.99 c = 5.34\) represents this situation. It’s also given that Davio bought twice as many pounds of celery as pounds of potatoes; therefore, \(c = 2 p\). Substituting \(2 p\) for \(c\) in the equation \(0.69 p + 0.99 c = 5.34\) yields \(0.69 p + 0.99(2 p)= 5.34\), which is equivalent to \(0.69 p + 1.98 p = 5.34\), or \(2.67 p = 5.34\). Dividing both sides of this equation by \(2.67\) yields \(p = 2\). Substituting \(2\) for \(p\) in the equation \(c = 2 p\) yields \(c = 2(2)\), or \(c = 4\). Therefore, Davio bought \(4\) pounds of celery. Choice A is incorrect. This is the number of pounds of potatoes, not the number of pounds of celery, Davio bought. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice B is correct. Any line in the xy-plane can be defined by an equation in the form y = mx + b, where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. From the graph, the y-intercept of the line is (0, 3). Therefore, b = 3. The slope of the line is the change in the value of y divided by the change in the value of x for any two points on the line. The line shown passes through (0, 3) and (1, 0), so, or m = –3. Therefore, the equation of the line is y = –3x + 3.Choices A and C are incorrect because the equations given in these choices represent a line with a positive slope. However, the line shown has a -slope. Choice D is incorrect because the equation given in this choice represents a line with slope of . However, the line shown has a slope of –3.

The correct answer is \(-32\). It’s given that the y-intercept of the graph of \(y = -6 x -32\) is \((0, y)\). Substituting \(0\) for \(x\) in this equation yields \(y = -6(0)-32\), or \(y = -32\). Therefore, the value of \(y\) that corresponds to the y-intercept of the graph of \(y = -6 x -32\) in the xy-plane is \(-32\).

Choice A is correct. It's given that \(68 x + 85 y = 1,258\) represents the situation where an interior designer earned a total of \($ 1,258\) last week from consulting for \(x\) hours and drawing up plans for \(y\) hours. Thus, \(68 x\) represents the amount earned, in dollars, from consulting for \(x\) hours, and \(85 y\) represents the amount earned, in dollars, from drawing up plans for \(y\) hours. Since \(68 x\) represents the amount earned, in dollars, from consulting for \(x\) hours, it follows that the interior designer earned \($ 68\) per hour consulting last week. Choice B is incorrect. The interior designer worked \(y\) hours, not \(68\) hours, drawing up plans last week. Choice C is incorrect. The interior designer earned \($ 85\) per hour, not \($ 68\) per hour, drawing up plans last week. Choice D is incorrect. The interior designer worked \(x\) hours, not \(68\) hours, consulting last week.

Choice C is correct. A line in the xy-plane can be represented by an equation of the form \(y = m x + b\), where \(m\) is the slope and \(b\) is the y-coordinate of the y-intercept of the line. It's given that line \(s\) passes through the point \((0, 0)\). Therefore, the y-coordinate of the y-intercept of line \(s\) is \(0\). It's also given that line \(s\) is parallel to the line represented by the equation \(y = 18 x + 2\). Since parallel lines have the same slope, it follows that the slope of line \(s\) is \(18\). Therefore, line \(s\) can be represented by the equation \(y = m x + b\), where \(m = 18\) and \(b = 0\). Substituting \(18\) for \(m\) and \(0\) for \(b\) in \(y = m x + b\) yields the equation \(y = 18 x + 0\), or \(y = 18 x\). If line \(s\) passes through the point \((4, d)\), then when \(x = 4\), \(y = d\) for the equation \(y = 18 x\). Substituting \(4\) for \(x\) and \(d\) for \(y\) in this equation yields \(d = 18(4)\), or \(d = 72\). Choice A is incorrect. This is the y-coordinate of the y-intercept of the line represented by the equation \(y = 18 x + 2\). Choice B is incorrect. This is the slope of the line represented by the equation \(y = 18 x + 2\). Choice D is incorrect. The line represented by the equation \(y = 18 x + 2\), not line \(s\), passes through the point \((4, 74)\).

Choice A is correct. Let \(x\) represent the number of rabbit snails that Naomi bought. It’s given that each rabbit snail costs \($ 8\). Therefore, the total cost, in dollars, of the rabbit snails that Naomi bought can be represented by the expression \(8 x\). It’s also given that each nerite snail costs \($ 6\), and that Naomi bought \(2\) nerite snails. Therefore, the total cost, in dollars, of the nerite snails that Naomi bought is \(6(2)\), or \(12\). Since Naomi bought both the rabbit snails and the nerite snails for a total of \($ 52\), the equation \(8 x + 12 = 52\) can be used to represent the situation. Subtracting \(12\) from both sides of this equation yields \(8 x = 40\). Dividing both sides of this equation by \(8\) yields \(x = 5\). Therefore, Naomi bought \(5\) rabbit snails. Choice B is incorrect. This is the total cost, in dollars, of the nerite snails that Naomi bought, not the number of rabbit snails. Choice C is incorrect. This is the cost, in dollars, of one rabbit snail and one nerite snail, not the number of rabbit snails that Naomi bought. Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. The equation representing the linear relationship shown can be written in slope-intercept form \(y = m x + b\), where \(m\) is the slope and \((0, b)\) is the y-intercept of the line. The line shown passes through the points \((0, 6)\) and \((2, 0)\). Given two points on a line, \((x_1, y_1)\) and \((x_2, y_2)\), the slope of the line can be calculated using the equation \(m = y_2 -y_1/x_2 -x_1\). Substituting \((0, 6)\) and \((2, 0)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in this equation yields \(m = -6/2 -0\), which is equivalent to \(m = -6 halves\), or \(m = -3\). Since \((0, 6)\) is the y-intercept, it follows that \(b = 6\). Substituting \(-3\) for \(m\) and \(6\) for \(b\) in the equation \(y = m x + b\) yields \(y = -3 x + 6\). Adding \(3 x\) to both sides of this equation yields \(3 x + y = 6\). Multiplying this equation by \(6\) yields \(18 x + 6 y = 36\). It follows that the equation \(18 x + R y = 36\), where \( R\) is a positive constant, represents this relationship. Choice A is incorrect. The graph of this relationship passes through the point \((0, 2)\), not \((0, 6)\). Choice B is incorrect. The graph of this relationship passes through the point \((0, 2)\), not \((0, 6)\). Choice D is incorrect. The graph of this relationship passes through the point \((-2, 0)\), not \((2, 0)\).

The correct answer is \(9\). It’s given that the equation \(y = p x + r\) defines the line. In this equation, \(p\) represents the slope of the line and \(r\) represents the y-coordinate of the y-intercept of the line. It’s given that the line has a slope of \(9\). Therefore, the value of \(p\) is \(9\).

The correct answer is \(14\). It's given that the equation \(46 = 2 a + 2 b\) gives the relationship between the side lengths \(a\) and \(b\) of a certain parallelogram. Substituting \(9\) for \(a\) in the given equation yields \(46 = 2(9)+ 2 b\), or \(46 = 18 + 2 b\). Subtracting \(18\) from both sides of this equation yields \(28 = 2 b\). Dividing both sides of this equation by \(2\) yields \(14 = b\). Therefore, if \(a = 9\), the value of \(b\) is \(14\).

Choice D is correct. The y-intercept of a line graphed in the xy-plane is the point where the line intersects the y-axis. The line graphed intersects the y-axis at the point \((0, 8)\). Therefore, the y-intercept of the line graphed is \((0, 8)\). Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors.

The correct answer is \(18/11\). For a line that passes through the points \((x_1, y_1)\) and \((x_2, y_2)\) in the xy-plane, the slope of the line can be calculated using the slope formula, \(m = y_2 -y_1/x_2 -x_1\). It's given that a line passes through the points \((4, 6)\) and \((15, 24)\) in the xy-plane. Substituting \((4, 6)\) for \((x_1, y_1)\) and \((15, 24)\) for \((x_2, y_2)\) in the slope formula, \(m = y_2 -y_1/x_2 -x_1\), yields \(m = 24 -6/15 -4\), or \(m = 18/11\). Therefore, the slope of the line is \(18/11\). Note that 18/11 and 1.636 are examples of ways to enter a correct answer.

The correct answer is \(55/4\). In the xy-plane, the graph of an equation in the form \(y = m x + b\), where \(m\) and \(b\) are constants, has a slope of \(m\) and a y-intercept of \((0, b)\). Applying the distributive property to the right-hand side of the given equation yields \(y = 27/4 x + 15/4 + 7 x\). Combining like terms yields \(y = 55/4 x + 15/4\). This equation is in the form \(y = m x + b\), where \(m = 55/4\) and \(b = 15/4\). It follows that the slope of the graph of \(y = 1 fourth(27 x + 15)+ 7 x\) in the xy-plane is \(55/4\). Note that 55/4 and 13.75 are examples of ways to enter a correct answer.

Choice C is correct. It's given that in triangle \( Q R S\), sides \( Q R\) and \( R S\) each have a length of \(x\) centimeters. Therefore, the expression \(2 x\) represents the sum of the lengths, in centimeters, of sides \( Q R\) and \( R S\). It's also given that side \( S Q\) has a length of \(y\) centimeters. Therefore, the expression \(2 x + y\) represents the sum of the lengths, in centimeters, of sides \( Q R\), \( R S\), and \( S Q\). Since \(2 x + y\) is the sum of the lengths, in centimeters, of the three sides of the triangle and \(2 x + y = 37\), it follows that \(37\) is the sum of the lengths, in centimeters, of the three sides of the triangle. Choice A is incorrect. The difference, in centimeters, between the lengths of sides \( Q R\) and \( S Q\) is \(x -y\), not \(37\). Choice B is incorrect. The difference, in centimeters, between the lengths of sides \( Q R\) and \( R S\) is \(x -x\), or \(0\), not \(37\). Choice D is incorrect. The length, in centimeters, of one of the two sides of equal length is \(x\), not \(37\).

The correct answer is 1.5. The total amount, in liters, of a saline solution can be expressed as the liters of each type of saline solution multiplied by the percent concentration of the saline solution. This gives,, and, where x is the amount, in liters, of 25% saline solution and 10%, 15%, and 25% are represented as 0.10, 0.15, and 0.25, respectively. Thus, the equation must be true. Multiplying 3 by 0.10 and distributing 0.15 to yields . Subtracting 0.15x and 0.30 from each side of the equation gives . Dividing each side of the equation by 0.10 yields . Note that 1.5 and 3/2 are examples of ways to enter a correct answer.

The correct answer is \(189/5\). A y-intercept of a graph in the xy-plane is a point where the graph intersects the y-axis, which is a point with an x-coordinate of \(0\). Substituting \(0\) for \(x\) in the given equation yields \(3(0)/7 = -5 y/9 + 21\), or \(0 = -5 y/9 + 21\). Subtracting \(21\) from both sides of this equation yields \(-21 = -5 y/9\). Multiplying both sides of this equation by \(-9\) yields \(189 = 5 y\). Dividing both sides of this equation by \(5\) yields \(189/5 = y\). Therefore, the y-coordinate of the y-intercept of the graph of the given equation in the xy-plane is \(189/5\). Note that 189/5 and 37.8 are examples of ways to enter a correct answer.

Choice A is correct. Since Jay walks at a speed of \(3\) miles per hour for \(w\) hours, Jay walks a total of \(3 w\) miles. Since Jay runs at a speed of \(5\) miles per hour for \(r\) hours, Jay runs a total of \(5 r\) miles. Therefore, the total number of miles Jay travels can be represented by \(3 w + 5 r\). Since the combined total number of miles is \(14\), the equation \(3 w + 5 r = 14\) represents this situation. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice A is correct. It's given that a neighborhood consists of a \(2\)-hectare park and a \(35\)-hectare residential area and that the total number of trees in the neighborhood is \(3,934\). It's also given that the equation \(2 x + 35 y = 3,934\) represents this situation. Since the total number of trees for a given area can be determined by taking the number of hectares times the average number of trees per hectare, this must mean that the terms \(2 x\) and \(35 y\) correspond to the number of trees in the park and in the residential area, respectively. Since \(2 x\) corresponds to the number of trees in the park, and \(2\) is the size of the park, in hectares, \(x\) must represent the average number of trees per hectare in the park. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice A is correct. The slope, \(m\), of a line in the xy-plane can be found using two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), and the slope formula \(m = y_2 -y_1/x_2 -x_1\). Based on the given table, the line representing the relationship between \(x\) and \(y\) in the xy-plane passes through the points \((-6, n + 184)\), \((-3, n + 92)\), and \((0, n)\), where \(n\) is a constant. Substituting two of these points, \((-3, n + 92)\) and \((0, n)\), for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the slope formula yields \(m = n -(n + 92)/0 -(-3)\), which is equivalent to \(m = n -n -92/0 + 3\), or \(m = -92/3\). Therefore, the slope of the line that represents this relationship in the xy-plane is \(-92/3\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. A line in the xy-plane with a slope of \(m\) and a y-intercept of \((0, b)\) can be represented by the equation \(y = m x + b\). It's given that the line has a slope of \(-1 half\). Therefore, \(m = -1 half\). It's also given that the line passes through the point \((0, 3)\). Therefore, \(b = 3\). Substituting \(-1 half\) for \(m\) and \(3\) for \(b\) in the equation \(y = m x + b\) yields \(y = -1 half x + 3\). Therefore, the equation \(y = -1 half x + 3\) represents this line. Choice A is incorrect. This equation represents a line in the xy-plane that passes through the point \((0, -3)\), not \((0, 3)\). Choice C is incorrect. This equation represents a line in the xy-plane that has a slope of \(1 half\), not \(-1 half\), and passes through the point \((0, -3)\), not \((0, 3)\). Choice D is incorrect. This equation represents a line in the xy-plane that has a slope of \(1 half\), not \(-1 half\).

Choice D is correct. It’s given that line j is perpendicular to line k and that line k is defined by the equation . This equation can be rewritten in slope-intercept form,, where m represents the slope of the line and b represents the y-coordinate of the y-intercept of the line, by subtracting x from both sides of the equation, which yields . Thus, the slope of line k is . Since line j and line k are perpendicular, their slopes are opposite reciprocals of each other. Thus, the slope of line j is 1. It’s given that the y-intercept of line j is . Therefore, the equation for line j in slope-intercept form is, which can be rewritten as .Choices A, B, and C are incorrect and may result from conceptual or calculation errors.

Choice B is correct. It’s given that is equal to the total amount of money, in dollars, a food truck charges for x drinks and y salads. Since each salad has the same price, it follows that the total charge for y salads is dollars. When, the value of the expression is, or 7.00. Therefore, the price for one salad is 7.00 dollars.Choice A is incorrect. Since each drink has the same price, it follows that the total charge for x drinks is dollars. Therefore, the price, in dollars, for one drink is 2.50, not 7.00. Choices C and D are incorrect. In the given equation, F represents the total charge, in dollars, when x drinks and y salads are bought at the food truck. No information is provided about the number of drinks or the number of salads that are bought during the day. Therefore, 7.00 doesn’t represent either of these quantities.

Choice C is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The line graphed intersects the y-axis at the point \((0, 5)\). Therefore, the y-intercept of the line graphed is \((0, 5)\). Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice C is correct. It's given from the graph that the points \((0, 7)\) and \((8, 0)\) lie on the line. For two points on a line, \((x_1, y_1)\) and \((x_2, y_2)\), the slope of the line can be calculated using the slope formula \(m = y_2 -y_1/x_2 -x_1\). Substituting \((0, 7)\) for \((x_1, y_1)\) and \((8, 0)\) for \((x_2, y_2)\) in this formula, the slope of the line can be calculated as \(m = -7/8 -0\), or \(m = -7 eighths\). It's also given that the point \((d, 4)\) lies on the line. Substituting \((d, 4)\) for \((x_1, y_1)\), \((8, 0)\) for \((x_2, y_2)\), and \(-7 eighths\) for \(m\) in the slope formula yields \(-7 eighths = -4/8 -d\), or \(-7 eighths = -4/8 -d\). Multiplying both sides of this equation by \(8 -d\) yields \(-7 eighths(8 -d)= -4\). Expanding the left-hand side of this equation yields \(-7 + 7 eighths d = -4\). Adding \(7\) to both sides of this equation yields \(7 eighths d = 3\). Multiplying both sides of this equation by \(8 sevenths\) yields \(d = 24/7\). Thus, the value of \(d\) is \(24/7\). Choice A is incorrect. This is the value of \(y\) when \(x = 4\). Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. In the xy-plane, the graph of an equation in the form \(y = m x + b\), where \(m\) and \(b\) are constants, has a slope of \(m\) and a y-intercept of \((0, b)\). Therefore, the y-intercept of the graph of \(y = 34 x + 81\) is \((0, 81)\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is 1.25. The y-coordinate of the x-intercept is 0, so 0 can be substituted for y, giving . This simplifies to . Multiplying both sides of by 5 gives . Dividing both sides of by 4 gives, which is equivalent to 1.25. Note that 1.25 and 5/4 are examples of ways to enter a correct answer.

Choice A is correct. Each of the given choices gives three values of \(x\): \(0\), \(2\), and \(4\). Substituting \(0\) for \(x\) in the given equation yields \(y = 70(0)+ 8\), or \(y = 8\). Therefore, when \(x = 0\), the corresponding value of \(y\) for the given equation is \(8\). Substituting \(2\) for \(x\) in the given equation yields \(y = 70(2)+ 8\), or \(y = 148\). Therefore, when \(x = 2\), the corresponding value of \(y\) for the given equation is \(148\). Substituting \(4\) for \(x\) in the given equation yields \(y = 70(4)+ 8\), or \(y = 288\). Therefore, when \(x = 4\), the corresponding value of \(y\) for the given equation is \(288\). Thus, if the three values of \(x\) are \(0\), \(2\), and \(4\), then their corresponding values of \(y\) are \(8\), \(148\), and \(288\), respectively, for the given equation. Choice B is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(y = 4 x + 70\). Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. The number of birds can be found by calculating the value of \(b\) when \(r = 16\) in the given equation. Substituting \(16\) for \(r\) in the given equation yields \(2.5 b + 5(16)= 80\), or \(2.5 b + 80 = 80\). Subtracting \(80\) from both sides of this equation yields \(2.5 b = 0\). Dividing both sides of this equation by \(2.5\) yields \(b = 0\). Therefore, if the business cares for \(16\) reptiles on a given day, it can care for \(0\) birds on this day. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It's given that the shipment consists of \(5\)-pound boxes and \(10\)-pound boxes with a total weight of \(220\) pounds. Let \(x\) represent the number of \(5\)-pound boxes and \(y\) represent the number of \(10\)-pound boxes in the shipment. Therefore, the equation \(5 x + 10 y = 220\) represents this situation. It's given that there are \(13\) \(10\)-pound boxes in the shipment. Substituting \(13\) for \(y\) in the equation \(5 x + 10 y = 220\) yields \(5 x + 10(13)= 220\), or \(5 x + 130 = 220\). Subtracting \(130\) from both sides of this equation yields \(5 x = 90\). Dividing both sides of this equation by \(5\) yields \(18\). Thus, there are \(18\) \(5\)-pound boxes in the shipment. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the number of \(10\)-pound boxes in the shipment.

Choice A is correct. It’s given that figure A and figure B (not shown) are both regular polygons and the sum of the perimeters of the two figures is \(63 inches\). It’s also given that \(x\) is the number of sides of figure A and \(y\) is the number of sides of figure B, and that the equation \(3 x + 6 y = 63\) represents this situation. Thus, \(3 x\) and \(6 y\) represent the perimeters, in inches, of figure A and figure B, respectively. Since \(6 y\) represents the perimeter, in inches, of figure B and \(y\) is the number of sides of figure B, it follows that each side of figure B has a length of \(6 inches\). Choice B is incorrect. The number of sides of figure B is \(y\), not \(6\). Choice C is incorrect. Since the perimeter, in inches, of figure A is represented by \(3 x\), each side of figure A has a length of \(3 inches\), not \(6 inches\). Choice D is incorrect. The number of sides of figure A is \(x\), not \(6\).

Choice D is correct. Since \(x\) represents the number of \(1\)-point questions and \(y\) represents the number of \(3\)-point questions, the assignment is worth a total of \(1 dot x + 3 dot y\), or \(x + 3 y\), points. Since the assignment is worth \(70\) points, the equation \(x + 3 y = 70\) represents this situation. Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors.

Choice B is correct. Avery earns $10 per hour working at job A. Therefore, if she works a hours at job A, she will earn dollars. Avery earns $20 per hour working at job B. Therefore, if she works b hours at job B, she will earn dollars. The graph shown represents all possible combinations of the number of hours Avery could have worked at the two jobs to earn s dollars. Therefore, if she worked a hours at job A, worked b hours at job B, and earned s dollars from both jobs, the following equation represents the graph:, where s is a constant. Identifying any point from the graph and substituting the values of the coordinates for a and b, respectively, in this equation yield the value of s. For example, the point, where and, lies on the graph. Substituting 16 for a and 0 for b in the equation yields, or . Similarly, the point, where and, lies on the graph. Substituting 0 for a and 8 for b in the equation yields, or .Choices A, C, and D are incorrect. If the value of s is 128, 200, or 320, then no points on the graph will satisfy this equation. For example, if the value of s is 128 (choice A), then the equation becomes . The point, where and, lies on the graph. However, substituting 16 for a and 0 for b in yields, or, which is false. Therefore, doesn’t satisfy the equation, and so the value of s can’t be 128. Similarly, if (choice C) or (choice D), then substituting 16 for a and 0 for b yields and, respectively, which are both false.

Choice A is correct. It's given that for a camπng trip a group bought \(x\) one-liter bottles of water and \(y\) three-liter bottles of water. Since the group bought \(x\) one-liter bottles of water, the total number of liters bought from \(x\) one-liter bottles of water is represented as \(1 x\), or \(x\). Since the group bought \(y\) three-liter bottles of water, the total number of liters bought from \(y\) three-liter bottles of water is represented as \(3 y\). It's given that the group bought a total of \(240\) liters; thus, the equation \(x + 3 y = 240\) represents this situation. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors. Choice D is incorrect. This equation represents a situation where the group bought \(x\) three-liter bottles of water and \(y\) one-liter bottles of water, for a total of \(240\) liters of water.

The correct answer is \(1 fourth\). It's given that line \(k\) is defined by \(y = 1 fourth x + 1\). It's also given that line \(j\) is parallel to line \(k\) in the xy-plane. A line in the xy-plane represented by an equation in slope-intercept form \(y = m x + b\) has a slope of \(m\) and a y-intercept of \((0, b)\). Therefore, the slope of line \(k\) is \(1 fourth\). Since parallel lines have equal slopes, the slope of line \(j\) is \(1 fourth\). Note that 1/4 and .25 are examples of ways to enter a correct answer.

The correct answer is \(5 thirteenths\). The graph of a line in the xy-plane can be represented by the equation \(y = m x + b\), where \(m\) is the slope of the line and \(b\) is the y-coordinate of the y-intercept. The given equation can be written as \(y =(5 thirteenths)x -23\). Therefore, the slope of the graph of this equation in the xy-plane is \(5 thirteenths\). Note that 5/13, .3846, 0.385, and 0.384 are examples of ways to enter a correct answer.

Choice B is correct. It’s given that the equation \(x + y = 1,440\) represents the number of minutes of daylight, \(x\), and the number of minutes of non-daylight, \(y\), on a particular day in Oak Park, Illinois. It’s also given that this day has \(670\) minutes of daylight. Substituting \(670\) for \(x\) in the equation \(x + y = 1,440\) yields \(670 + y = 1,440\). Subtracting \(670\) from both sides of this equation yields \(y = 770\). Therefore, this day has \(770\) minutes of non-daylight. Choice A is incorrect. This is the number of minutes of daylight, not non-daylight, on this day. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This is the total number of minutes of daylight and non-daylight.

The correct answer is \(40\). It's given that \(5 G + 45 R = 380\), where \( G\) is the number of green tokens and \( R\) is the number of red tokens won by one student and these tokens are worth a total of \(380\) points. Since the equation represents the situation where the student won points with green tokens and red tokens for a total of \(380\) points, each term on the left-hand side of the equation represents the number of points won for one of the colors. Since the coefficient of \( G\) in the given equation is \(5\), a green token must be worth \(5\) points. Similarly, since the coefficient of \( R\) in the given equation is \(45\), a red token must be worth \(45\) points. Therefore, a red token is worth \(45 -5\) points, or \(40\) points, more than a green token.

Choice B is correct. Two lines are perpendicular if their slopes are -reciprocals, meaning that the slope of the first line is equal to \(-1\) divided by the slope of the second line. Each equation in the given pair of equations can be written in slope-intercept form, \(y = m x + b\), where \(m\) is the slope of the graph of the equation in the xy-plane and \((0, b)\) is the y-intercept. For the first equation, \(5 x + 7 y = 1\), subtracting \(5 x\) from both sides gives \(7 y = -5 x + 1\), and dividing both sides of this equation by \(7\) gives \(y = -5 sevenths x + 1 seventh\). Therefore, the slope of the graph of this equation is \(-5 sevenths\). For the second equation, \(a x + b y = 1\), subtracting \(a x\) from both sides gives \(b y = -a x + 1\), and dividing both sides of this equation by \(b\) gives \(y = -a/b x + 1/b\). Therefore, the slope of the graph of this equation is \(-a/b\). Since the graph of the given pair of equations is a pair of perpendicular lines, the slope of the graph of the second equation, \(-a/b\), must be the -reciprocal of the slope of the graph of the first equation, \(-5 sevenths\). The -reciprocal of \(-5 sevenths\) is \( -1/(-5 sevenths)\), or \(7 fifths\). Therefore, \(-a/b = 7 fifths\), or \(a/b = -7 fifths\). Similarly, rewriting the equations in choice B in slope-intercept form yields \(y = -10/7 x + 1 seventh\) and \(y = -a/2 b x + 1/2 b\). It follows that the slope of the graph of the first equation in choice B is \(-10/7\) and the slope of the graph of the second equation in choice B is \(-a/2 b\). Since \(a/b = -7 fifths\), \(-a/2 b\) is equal to \(-(1 half)(-7 fifths)\), or \(7 tenths\). Since \(7 tenths\) is the -reciprocal of \(-10/7\), the pair of equations in choice B represents a pair of perpendicular lines. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. It’s given that the weight of the granite used in the garden is 330 pounds. The weight of the limestone used in the garden is a product of its weight per volume, in lb/ft3, and its volume, in ft3. Therefore, the weight of the limestone used in the garden can be represented by, where x is the volume, in ft3, of the limestone used. Similarly, the weight of the basalt used in the garden can be represented by, where y is the volume, in ft3, of the basalt used. It’s given that the total weight of the rocks used in the garden will be 1,000 pounds. Thus, the sum of the weights of the three rock types used is 1,000 pounds, which can be represented by the equation . Subtracting 330 from both sides of this equation yields .Choice A is incorrect. This equation uses the weight per volume of granite instead of limestone. Choice B is incorrect. This equation uses the weight per volume of granite instead of basalt, and doesn’t take into account the 330 pounds of granite that will be used in the garden. Choice D is incorrect. This equation doesn’t take into account the 330 pounds of granite that will be used in the garden.

Choice A is correct. Each of the choices is a linear equation in the form y = mx + b, where m and b are constants. In this equation, m represents the change in y for each increase in x by 1. From the table, it can be determined that the value of y increases by 2 for each increase in x by 1. In other words, for the pairs of x and y in the given table, m = 2. The value of b can be found by substituting the values of x and y from any row of the table and substituting the value of m into the equation y = mx + b and then solving for b. For example, using x = 1, y = 5, and m = 2 yields 5 = 2(1) + b. Solving for b yields b = 3. Therefore, the equation y = 2x + 3 could represent the relationship between x and y in the given table.Alternatively, if an equation represents the relationship between x and y, then when each pair of x and y from the table is substituted into the equation, the result will be a true statement. Of the equations given, the equation y = 2x + 3 in choice A is the only equation that results in a true statement when each of the pairs of x and y are substituted into the equation.Choices B, C, and D are incorrect because when at least one pair of x and y from the table is substituted into the equations given in these choices, the result is a false statement. For example, when the pair x = 4 and y = 11 is substituted into the equation in choice B, the result is 11 = 3(4) – 2, or 11 = 10, which is false.

Choice D is correct. It’s given that line \(h\) is defined by \(1 fifth x + 1 seventh y -70 = 0\). This equation can be written in slope-intercept form \(y = m x + b\), where \(m\) is the slope of line \(h\) and \(b\) is the y-coordinate of the y-intercept of line \(h\). Adding \(70\) to both sides of \(1 fifth x + 1 seventh y -70 = 0\) yields \(1 fifth x + 1 seventh y = 70\). Subtracting \(1 fifth x\) from both sides of this equation yields \(1 seventh y = -1 fifth x + 70\). Multiplying both sides of this equation by \(7\) yields \(y = -7 fifths x + 490\). Therefore, the slope of line \(h\) is \(-7 fifths\). It’s given that line \(j\) is perpendicular to line \(h\) in the xy-plane. Two lines are perpendicular if their slopes are -reciprocals, meaning that the slope of the first line is equal to \(-1\) divided by the slope of the second line. Therefore, the slope of line \(j\) is the -reciprocal of the slope of line \(h\). The -reciprocal of \(-7 fifths\) is \( -1/(-7 fifths)\), or \(5 sevenths\). Therefore, the slope of line \(j\) is \(5 sevenths\). Choice A is incorrect. This is the slope of a line in the xy-plane that is parallel, not perpendicular, to line \(h\). Choice B is incorrect. This is the reciprocal, not the -reciprocal, of \(-7 fifths\). Choice C is incorrect. This is the -, not the -reciprocal, of \(-7 fifths\).

Choice D is correct. It’s given that the graph shows the linear relationship between \(x\) and \(y\). The given graph passes through the points \((0, -5)\), \((1, -3)\), and \((2, -1)\). It follows that when \(x = 0\), the corresponding value of \(y\) is \(-5\), when \(x = 1\), the corresponding value of \(y\) is \(-3\), and when \(x = 2\), the corresponding value of \(y\) is \(-1\). Of the given choices, only the table in choice D gives these three values of \(x\) and their corresponding values of \(y\) for the relationship shown in the graph. Choice A is incorrect. This table represents a relationship between \(x\) and \(y\) such that the graph passes through the points \((0, 0)\), \((1, -7)\), and \((2, -9)\). Choice B is incorrect. This table represents a relationship between \(x\) and \(y\) such that the graph passes through the points \((0, 0)\), \((1, -3)\), and \((2, -1)\). Choice C is incorrect. This table represents a linear relationship between \(x\) and \(y\) such that the graph passes through the points \((0, -5)\), \((1, -7)\), and \((2, -9)\).

The correct answer is \(41\). The number of cupcakes Vivian bought can be found by first finding the amount Vivian spent on cupcakes. The amount Vivian spent on cupcakes can be found by subtracting the amount Vivian spent on party hats from the total amount Vivian spent. The amount Vivian spent on party hats can be found by multiplying the cost per package of party hats by the number of packages of party hats, which yields \($ 3 dot 10\), or \($ 30\). Subtracting the amount Vivian spent on party hats, \($ 30\), from the total amount Vivian spent, \($ 71\), yields \($ 71 -$ 30\), or \($ 41\). Since the amount Vivian spent on cupcakes was \($ 41\) and each cupcake cost \($ 1\), it follows that Vivian bought \(41\) cupcakes.

Choice D is correct. The equation of a line in the xy-plane can be written as \(y = m x + b\), where \(m\) represents the slope of the line and \((0, b)\) represents the y-intercept of the line. It's given that the slope of the line is \(1 ninth\). It follows that \(m = 1 ninth\). It's also given that the line passes through the point \((0, 14)\). It follows that \(b = 14\). Substituting \(1 ninth\) for \(m\) and \(14\) for \(b\) in \(y = m x + b\) yields \(y = 1 ninth x + 14\). Thus, the equation \(y = 1 ninth x + 14\) represents this line. Choice A is incorrect. This equation represents a line with a slope of \(-1 ninth\) and a y-intercept of \((0, -14)\). Choice B is incorrect. This equation represents a line with a slope of \(-1 ninth\) and a y-intercept of \((0, 14)\). Choice C is incorrect. This equation represents a line with a slope of \(1 ninth\) and a y-intercept of \((0, -14)\).

The correct answer is 30. The slope of a line can be found by using the slope formula, . It’s given that the slope of the line is 2; therefore, . According to the table, the points and lie on the line. Substituting the coordinates of these points into the equation gives . Multiplying both sides of this equation by gives, or . Solving for k gives . According to the table, the points and also lie on the line. Substituting the coordinates of these points into gives . Solving for n gives . Therefore,, or 30.

Choice D is correct. An equation defining a line in the xy-plane can be written in the form \(y = m x + b\), where \(m\) represents the slope and \((0, b)\) represents the y-intercept of the line. It’s given that line \(k\) passes through the point \((0, -5)\); therefore, \(b = -5\). The slope, \(m\), of a line can be found using any two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), and the slope formula \(m = y_2 -y_1/x_2 -x_1\). Substituting the points \((0, -5)\) and \((1, -1)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the slope formula yields \(m = (-1 -(-5))/(1 -0)\), or \(m = 4\). Substituting \(4\) for \(m\) and \(-5\) for \(b\) in the equation \(y = m x + b\) yields \(y = 4 x -5\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice C is correct. It's given that when line \(n\) is graphed in the xy-plane, it has an x-intercept of \((-4, 0)\) and a y-intercept of \((0, 86/3)\). The slope, \(m\), of a line can be found using any two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), and the slope formula \(m = y_2 -y_1/x_2 -x_1\). Substituting the points \((-4, 0)\) and \((0, 86/3)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the slope formula yields \(m = \86/3 -0/0 -(-4)\), or \(m = 43/6\). Therefore, the slope of line \(n\) is \(43/6\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the slope of a line that has an x-intercept of \((86/3, 0)\) and a y-intercept of \((0, -4)\). Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is \(15\). It's given that the equation \(46 = 2 x + 2 y\) gives the perimeter of a rectangular rug that has length \(x\), in feet, and width \(y\), in feet. It's also given that the width of the rug is \(8\) feet. Substituting \(8\) for \(y\) in the equation \(46 = 2 x + 2 y\) yields \(46 = 2 x + 2(8)\), or \(46 = 2 x + 16\). Subtracting \(16\) from both sides of this equation yields \(30 = 2 x\). Dividing both sides of this equation by \(2\) yields \(15 = x\). Since \(x\) represents the length, in feet, of the rug, it follows that the length of the rug is \(15\) feet.

Choice A is correct. The table shows the cost of limestone is $0.03/lb, and the weight per volume for limestone is 120 lb/ft3. Therefore, the term represents the cost, in dollars, of w ft3 of limestone. Similarly, the term represents the cost, in dollars, of z ft3 of basalt. The given equation shows that the total cost of all the rocks used in the project is $7,576.20. Since it’s given that all four rock types are used in the project, the remaining term, 3,385.80, represents the cost, in dollars, of the granite and sandstone needed for the project.Choice B is incorrect. The cost of basalt and limestone needed for the project can be represented by . Choice C is incorrect. The cost of the basalt needed for the project can be represented by the expression . Choice D is incorrect and may result from neglecting to include granite in the rock types used for the project.

The correct answer is \(59/9\). When the graph of an equation in the form \(A x + B y = C\), where \(A\), \(B\), and \(C\) are constants, is translated down \(k\) units in the xy-plane, the resulting graph can be represented by the equation \(A x + B(y + k)= C\). It’s given that the graph of \(9 x -10 y = 19\) is translated down \(4\) units in the xy-plane. Therefore, the resulting graph can be represented by the equation \(9 x -10(y + 4)= 19\), or \(9 x -10 y -40 = 19\). Adding \(40\) to both sides of this equation yields \(9 x -10 y = 59\). The x-coordinate of the x-intercept of the graph of an equation in the xy-plane is the value of \(x\) in the equation when \(y = 0\). Substituting \(0\) for \(y\) in the equation \(9 x -10 y = 59\) yields \(9 x -10(0)= 59\), or \(9 x = 59\). Dividing both sides of this equation by \(9\) yields \(x = 59/9\). Therefore, the x-coordinate of the x-intercept of the resulting graph is \(59/9\). Note that 59/9, 6.555, and 6.556 are examples of ways to enter a correct answer.

Choice C is correct. It’s given that a total of \(2\) squares each have side length \(r\). Therefore, each of the squares has perimeter \(4 r\). Since there are a total of \(2\) squares, the sum of the perimeters of these squares is \(4 r + 4 r\), which is equivalent to \(2(4 r)\), or \(8 r\). It’s also given that a total of \(6\) equilateral triangles each have side length \(t\). Therefore, each of the equilateral triangles has perimeter \(3 t\). Since there are a total of \(6\) equilateral triangles, the sum of the perimeters of these triangles is \(3 t + 3 t + 3 t + 3 t + 3 t + 3 t\), which is equivalent to \(6(3 t)\), or \(18 t\). Since the sum of the perimeters of the squares is \(8 r\) and the sum of the perimeters of the triangles is \(18 t\), the sum of the perimeters of all these squares and triangles is \(8 r + 18 t\). It’s given that the sum of the perimeters of all these squares and triangles is \(210\). Therefore, the equation \(8 r + 18 t = 210\) represents this situation. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. Since a 160-pound adult uses 200 calories for every 30 minutes of hiking, then the same adult uses calories after hiking for h 30-minute periods. Similarly, the same adult uses calories after bicycling for b 30-minute periods. Therefore, the equation represents the situation where a 160-pound adult uses a total of 1,900 calories from hiking for h 30-minute periods and bicycling for b 30-minute periods. It’s given that the adult completes 1 hour, or 2 30-minute periods, of bicycling. Substituting 2 for b in the equation yields, or . Subtracting 300 from both sides of this equation yields . Dividing both sides by 200 yields . Since h represents the number of 30-minute periods spent hiking and there are 2 30-minute periods in every hour, it follows that the adult will need to hike for, or 4 hours to use a total of 1,900 calories from bicycling and hiking.Choice A is incorrect and may result from solving the equation . This represents 0 30-minute periods bicycling instead of 2. Choice B is incorrect and may result from solving the equation . This represents 1 30-minute period of bicycling instead of 2. Choice C is incorrect. This may result from determining that the number of 30-minute periods the adult should hike is 8, but then subtracting 2 from 8, rather than dividing 8 by 2, to find the number of hours the adult should hike.

Choice D is correct. Since a 10-ride pass costs $12.50 and g is the number of 10-ride passes Tony buys in a month, the expression represents the amount Tony spends on 10-ride passes in a month. Since a single-ride pass costs $1.50 and t is the number of single-ride passes Tony buys in a month, the expression represents the amount Tony spends on single-ride passes in a month. Therefore, the sum represents the amount he spends on the two types of passes in a month. Since Tony spends a total of $80 on passes in a month, this expression can be set equal to 80, producing .Choices A and B are incorrect. The expression represents the total number of the two types of passes Tony buys in a month, not the amount Tony spends, which is equal to 80, nor the cost of one of each pass, which is equal to . Choice C is incorrect and may result from reversing the cost for each type of pass Tony buys in a month.

Choice B is correct. It's given that line \(k\) is defined by \(y = 7 x + 1 eighth\). For an equation in slope-intercept form \(y = m x + b\), \(m\) represents the slope of the line defined by this equation in the xy-plane and \(b\) represents the y-coordinate of the y-intercept of this line. Therefore, the slope of line \(k\) is \(7\). It’s also given that line \(j\) is perpendicular to line \(k\) in the xy-plane. Therefore, the slope of line \(j\) is the opposite reciprocal of the slope of line \(k\). The opposite reciprocal of \(7\) is \(-1 seventh\). Therefore, the slope of line \(j\) is \(-1 seventh\). Choice A is incorrect. This is the opposite reciprocal of the y-coordinate of the y-intercept, not the slope, of line \(k\). Choice C is incorrect. This is the y-coordinate of the y-intercept of line \(k\), not the slope of line \(j\). Choice D is incorrect. This is the slope of a line that is parallel, not perpendicular, to line \(k\).

Choice A is correct. It is given that line is perpendicular to a line whose equation is x = 2. A line whose equation is a constant value of x is vertical, so must therefore be horizontal. Horizontal lines have a slope of 0, so has a slope of 0.Choice B is incorrect. A line with slope is perpendicular to a line with slope 2. However, the line with equation x = 2 is vertical and has undefined slope (not slope of 2). Choice C is incorrect. A line with slope –2 is perpendicular to a line with slope . However, the line with equation x = 2 has undefined slope (not slope of ). Choice D is incorrect; this is the slope of the line x = 2 itself, not the slope of a line perpendicular to it.

Choice D is correct. A linear relationship can be represented by an equation of the form \(y = m x + b\), where \(m\) and \(b\) are constants. It’s given in the table that when \(x = 0\), \(y = 18\). Substituting \(0\) for \(x\) and \(18\) for \(y\) in \(y = m x + b\) yields \(18 = m(0)+ b\), or \(18 = b\). Substituting \(18\) for \(b\) in the equation \(y = m x + b\) yields \(y = m x + 18\). It’s also given in the table that when \(x = 1\), \(y = 13\). Substituting \(1\) for \(x\) and \(13\) for \(y\) in the equation \(y = m x + 18\) yields \(13 = m(1)+ 18\), or \(13 = m + 18\). Subtracting \(18\) from both sides of this equation yields \(-5 = m\). Therefore, the equation \(y = -5 x + 18\) represents the relationship between \(x\) and \(y\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice B is correct. It's given that the equation \(40 x + 20 y = 160\) represents the number of sweaters, \(x\), and the number of shirts, \(y\), that Yesenia purchased for \($ 160\). If Yesenia purchased \(2\) sweaters, the number of shirts she purchased can be calculated by substituting \(2\) for \(x\) in the given equation, which yields \(40(2)+ 20 y = 160\), or \(80 + 20 y = 160\). Subtracting \(80\) from both sides of this equation yields \(20 y = 80\). Dividing both sides of this equation by \(20\) yields \(y = 4\). Therefore, if Yesenia purchased \(2\) sweaters, she purchased \(4\) shirts. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the number of shirts Yesenia purchased if she purchased \(0\) sweaters. Choice D is incorrect. This is the price, in dollars, for each sweater, not the number of shirts Yesenia purchased.

Choice C is correct. The relationship between s and P can be modeled by a linear equation of the form P = ks + a, where k and a are constants. The table shows that P increases by 10 when s increases by 3, so k = . To solve for a, substitute one of the given pairs of values for s and P: when s = 3, P = 8, so, which yields a = –2. The solution is therefore .Choice A is incorrect. When s = 3, P = 8, but 3(3) + 10 = 19 ≠︀ 8. Choice B is incorrect. This may result from using the first number given for P in the table as the constant term a in the linear equation P = ks + a, which is true only when s = 0. Choice D is incorrect and may result from using the reciprocal of the slope of the line.

Choice A is correct. The slope-intercept form of an equation for a line is, where m is the slope of the line and b is the y-coordinate of the y-intercept of the line. It’s given that the slope is 6, so . It’s also given that the line passes through the point on the y-axis, so . Substituting and into the equation gives .Choices B, C, and D are incorrect and may result from misinterpreting the slope-intercept form of an equation of a line.

Choice B is correct. Since Fertilizer A contains 60% filler materials by weight, it follows that x pounds of Fertilizer A consists of 0.6x pounds of filler materials. Similarly, y pounds of Fertilizer B consists of 0.4y pounds of filler materials. When x pounds of Fertilizer A and y pounds of Fertilizer B are combined, the result is 240 pounds of filler materials. Therefore, the total amount, in pounds, of filler materials in a mixture of x pounds of Fertilizer A and y pounds of Fertilizer B can be expressed as .Choice A is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B. Fertilizer A consists of 0.6x pounds of filler materials and Fertilizer B consists of 0.4y pounds of filler materials. Therefore, is equal to 240, not . Choice C is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B and incorrectly represents how to take the percentage of a value mathematically. Choice D is incorrect. This choice incorrectly represents how to take the percentage of a value mathematically. Fertilizer A consists of 0.6x pounds of filler materials, not 60x pounds of filler materials, and Fertilizer B consists of 0.4y pounds of filler materials, not 40y pounds of filler materials.

Choice D is correct. Since \(x\) represents the number of \(1\)-minute segments and \(y\) represents the number of \(3\)-minute segments, the total length of the video is \(1 dot x + 3 dot y\), or \(x + 3 y\), minutes. Since the video is \(70\) minutes long, the equation \(x + 3 y = 70\) represents this situation. Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors.

Choice C is correct. Since Mario purchased 4 binders that cost x dollars each, the expression represents the total cost, in dollars, of the 4 binders he purchased. Since Mario purchased 3 notebooks that cost y dollars each, the expression represents the total cost, in dollars, of the 3 notebooks he purchased. Therefore, the expression represents the total cost, in dollars, for all binders and notebooks he purchased. In the given equation, the expression is equal to 24. Therefore, it follows that 24 is the total cost, in dollars, for all binders and notebooks purchased.Choice A is incorrect. This is represented by the expression in the given equation. Choice B is incorrect. This is represented by the expression in the given equation. Choice D is incorrect. This is represented by the expression .

Choice B is correct. The graph shown is a line passing through the points \((0, 40)\) and \((60, 0)\). Since the relationship between \(x\) and \(y\) is linear, if two points on the graph make a linear equation true, then the equation represents the relationship. Substituting \(0\) for \(x\) and \(40\) for \(y\) in the equation in choice B, \(8 x + 12 y = 480\), yields \(8(0)+ 12(40)= 480\), or \(480 = 480\), which is true. Substituting \(60\) for \(x\) and \(0\) for \(y\) in the equation \(8 x + 12 y = 480\) yields \(8(60)+ 12(0)= 480\), or \(480 = 480\), which is true. Therefore, the equation \(8 x + 12 y = 480\) represents the relationship between \(x\) and \(y\). Choice A is incorrect. The point \((0, 40)\) is not on the graph of this equation, since \(40 = 8(0)+ 12\), or \(40 = 12\), is not true. Choice C is incorrect. The point \((0, 40)\) is not on the graph of this equation, since \(40 = 12(0)+ 8\), or \(40 = 8\), is not true. Choice D is incorrect. The point \((0, 40)\) is not on the graph of this equation, since \(12(0)+ 8(40)= 480\), or \(320 = 480\), is not true.

Choice D is correct. The x-coordinate \(a\) of the x-intercept \((a, 0)\) can be found by substituting \(0\) for \(y\) in the given equation, which gives \(7 x + 2(0)= -31\), or \(7 x = -31\). Dividing both sides of this equation by \(7\) yields \(x = -31/7\). Therefore, the value of \(a\) is \(-31/7\). The y-coordinate \(b\) of the y-intercept \((0, b)\) can be found by substituting \(0\) for \(x\) in the given equation, which gives \(7(0)+ 2 y = -31\), or \(2 y = -31\). Dividing both sides of this equation by \(2\) yields \(y = -31/2\). Therefore, the value of \(b\) is \(-31/2\). It follows that the value of \(b/a\) is \( -31/2/ -31/7\), which is equivalent to \((31/2)(7 30 firsts)\), or \(7 halves\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice B is correct. The linear relationship between \(x\) and \(y\) can be represented by an equation of the form \(y -y_1 = m(x -x_1)\), where \(m\) is the slope of the graph of the equation in the xy-plane and \((x_1, y_1)\) is a point on the graph. The slope of a line can be found using two points on the line and the slope formula \(m = y_2 -y_1/x_2 -x_1\). Each value of \(x\) and its corresponding value of \(y\) in the table can be represented by a point \((x, y)\). Substituting the points \((-s, 21)\) and \((s, 15)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the slope formula yields \(m = 15 -21/s -(-s)\), which gives \(m = -6/2 s\), or \(m = -3/s\). Substituting \(-3/s\) for \(m\) and the point \((s, 15)\) for \((x_1, y_1)\) in the equation \(y -y_1 = m(x -x_1)\) yields \(y -15 = -3/s(x -s)\). Distributing \(-3/s\) on the right-hand side of this equation yields \(y -15 = -3 x/s + 3\). Adding \(15\) to each side of this equation yields \(y = -3 x/s + 18\). Multiplying each side of this equation by \(s\) yields \(s y = -3 x + 18 s\). Adding \(3 x\) to each side of this equation yields \(3 x + s y = 18 s\). Therefore, the equation \(3 x + s y = 18 s\) represents this relationship. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It’s given that the food truck buys forks for \($ 0.04\) each. Therefore, the cost, in dollars, of \(x\) forks can be represented by the expression \(0.04 x\). It’s also given that the food truck buys plates for \($ 0.48\) each. Therefore, the cost, in dollars, of \(y\) plates can be represented by the expression \(0.48 y\). Since the total cost of \(x\) forks and \(y\) plates is \($ 661.76\), the equation \(0.04 x + 0.48 y = 661.76\) represents this situation. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This equation represents a situation in which the food truck buys forks for \($ 0.48\) each and plates for \($ 0.04\) each.

The correct answer is 1.5. The point corresponds to the situation where 16 cornflowers and 0 wallflowers are purchased. Since the total spent on the two types of flowers is $24.00, it follows that the price of 16 cornflowers is $24.00, and the price of one cornflower is $1.50. Note that 1.5 and 3/2 are examples of ways to enter a correct answer.

Choice C is correct. Each of the given choices gives three values of \(x\): \(0\), \(1\), and \(2\). Substituting \(0\) for \(x\) in the given equation yields \(y = -4(0)+ 40\), or \(y = 40\). Therefore, when \(x = 0\), the corresponding value of \(y\) for the given equation is \(40\). Substituting \(1\) for \(x\) in the given equation yields \(y = -4(1)+ 40\), or \(y = 36\). Therefore, when \(x = 1\), the corresponding value of \(y\) for the given equation is \(36\). Substituting \(2\) for \(x\) in the given equation yields \(y = -4(2)+ 40\), or \(y = 32\). Therefore, when \(x = 2\), the corresponding value of \(y\) for the given equation is \(32\). Choice C gives three values of \(x\), \(0\), \(1\), and \(2\), and their corresponding values of \(y\), \(40\), \(36\), and \(32\), respectively, for the given equation. Choice A is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(y = -4 x\). Choice B is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(y = 4 x + 40\). Choice D is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(y = 4 x\).

Choice C is correct. The given equation can be rewritten in slope-intercept form,, where m represents the slope of the line represented by the equation, and k represents the y-coordinate of the y-intercept of the line. Subtracting ax from both sides of the equation yields, and dividing both sides of this equation by b yields, or . With the equation now in slope-intercept form, it shows that, which means the y-coordinate of the y-intercept is 1. It’s given that a and b are both greater than 0 (positive) and that . Since, the slope of the line must be a value between and 0. Choice C is the only graph of a line that has a y-value of the y-intercept that is 1 and a slope that is between and 0.Choices A, B, and D are incorrect because the slopes of the lines in these graphs aren’t between and 0.

Choice B is correct. Of the city’s total expense budget for one year, the city budgeted y million dollars for departmental expenses and 201 million dollars for all other expenses. This means that the expression represents the total expense budget, in millions of dollars, for one year. It’s given that the total expense budget for one year is x million dollars. It follows then that the expression is equivalent to x, or . Subtracting y from both sides of this equation yields . By the symmetric property of equality, this is the same as .Choices A and C are incorrect. Because it’s given that the total expense budget for one year, x million dollars, is comprised of the departmental expenses, y million dollars, and all other expenses, 201 million dollars, the expressions and both must be equivalent to a value greater than 201 million dollars. Therefore, the equations and aren’t true. Choice D is incorrect. The value of x must be greater than the value of y. Therefore, can’t represent this relationship.

Choice C is correct. An equation of a line can be written in the form \(y = m x + b\), where \(m\) is the slope of the line and \((0, b)\) is the y-intercept of the line. The line shown passes through the point \((0, -8)\), so \(b = -8\). The line shown also passes through the point \((-8, 0)\). The slope, \(m\), of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the equation \(m = y_2 -y_1/x_2 -x_1\). For the points \((0, -8)\) and \((-8, 0)\), this gives \(m = (-8)-0/0 -(-8)\), or \(m = -1\). Substituting \(-8\) for \(b\) and \(-1\) for \(m\) in \(y = m x + b\) yields \(y =(-1)x +(-8)\), or \(y = -x -8\). Therefore, an equation of the graph shown is \(y = -x -8\). Choice A is incorrect. This is an equation of a line with a slope of \(-2\), not \(-1\). Choice B is incorrect. This is an equation of a line with a slope of \(1\), not \(-1\). Choice D is incorrect. This is an equation of a line with a slope of \(2\), not \(-1\).

Choice B is correct. It's given that the graph shows the possible combinations of the number of pounds of tangerines, \(x\), and the number of pounds of lemons, \(y\), that could be purchased for \($ 18\) at a certain store. If Melvin purchased lemons and \(4\) pounds of tangerines for a total of \($ 18\), the number of pounds of lemons he purchased is represented by the y-coordinate of the point on the graph where \(x = 4\). For the graph shown, when \(x = 4\), \(y = 10\). Therefore, if Melvin purchased lemons and \(4\) pounds of tangerines for a total of \($ 18\), then he purchased \(10\) pounds of lemons. Choice A is incorrect. This is the number of pounds of tangerines Melvin purchased if he purchased tangerines and \(4\) pounds of lemons for a total of \($ 18\). Choice C is incorrect. This is the number of pounds of lemons Melvin purchased if he purchased lemons and \(2\) pounds of tangerines for a total of \($ 18\). Choice D is incorrect. This is the number of pounds of lemons Melvin purchased if he purchased lemons and \(1\) pound of tangerines for a total of \($ 18\).

Choice C is correct. It’s given that the equation \(3 x + 5 y = 32\) represents the situation where Keenan filled \(x\) small jars and \(y\) large jars with all the vegetable broth he made, which was \(32\) cups. Therefore, \(3 x\) represents the total number of cups of vegetable broth in the small jars and \(5 y\) represents the total number of cups of vegetable broth in the large jars. Choice A is incorrect. The number of large jars Keenan filled is represented by \(y\), not \(5 y\). Choice B is incorrect. The number of small jars Keenan filled is represented by \(x\), not \(5 y\). Choice D is incorrect. The total number of cups of vegetable broth in the small jars is represented by \(3 x\), not \(5 y\).

Choice A is correct. An equation for the graph shown can be written in slope-intercept form \(y = m x + b\), where \(m\) is the slope of the graph and its y-intercept is \((0, b)\). Since the y-intercept of the graph shown is \((0, 2)\), the value of \(b\) is \(2\). Since the graph also passes through the point \((4, 1)\), the slope can be calculated as \(1 -2/4 -0\), or \(-1 fourth\). Therefore, the value of \(m\) is \(-1 fourth\). Substituting \(-1 fourth\) for \(m\) and \(2\) for \(b\) in the equation \(y = m x + b\) yields \(y = -1 fourth x + 2\). It’s given that an equation for the graph shown is \(y = f(x)+ 14\). Substituting \(f(x)+ 14\) for \(y\) in the equation \(y = -1 fourth x + 2\) yields \(f(x)+ 14 = -1 fourth x + 2\). Subtracting \(14\) from both sides of this equation yields \(f(x)= -1 fourth x -12\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is \(44/3\). A linear equation can be written in the form \(y = m x + b\), where \(m\) is the slope of the graph of the equation in the xy-plane and \((0, b)\) is the y-intercept. Distributing the \(1 third\) in the equation \(y = 1 third(29 x + 10)+ 5 x\) yields \(y = 29/3 x + 10/3 + 5 x\). Combining like terms on the right-hand side of this equation yields \(y = 44/3 x + 10/3\). This equation is in the form \(y = m x + b\), where \(m = 44/3\) and \(b = 10/3\). Therefore, the slope of the graph of the given equation in the xy-plane is \(44/3\). Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.

The correct answer is \(24\). A line in the xy-plane can be defined by the equation \(y = m x + b\), where \(m\) is the slope of the line and \(b\) is the y-coordinate of the y-intercept of the line. It's given that line \(ell\) passes through the point \((0, 0)\). Therefore, the y-coordinate of the y-intercept of line \(ell\) is \(0\). It's given that line \(ell\) is parallel to the line represented by the equation \(y = 8 x + 2\). Since parallel lines have the same slope, it follows that the slope of line \(ell\) is \(8\). Therefore, line \(ell\) can be defined by an equation in the form \(y = m x + b\), where \(m = 8\) and \(b = 0\). Substituting \(8\) for \(m\) and \(0\) for \(b\) in \(y = m x + b\) yields the equation \(y = 8 x + 0\), or \(y = 8 x\) . If line \(ell\) passes through the point \((3, d)\), then when \(x = 3\), \(y = d\) for the equation \(y = 8 x\). Substituting \(3\) for \(x\) and \(d\) for \(y\) in the equation \(y = 8 x\) yields \(d = 8(3)\), or \(d = 24\).

The correct answer is \(4\). It's given that line \(k\) is parallel to line \(j\). It follows that the slope of line \(k\) is equal to the slope of line \(j\). Given two points on a line in the xy-plane, \((x_1, y_1)\) and \((x_2, y_2)\), the slope of the line can be calculated as \(y_2 -y_1/x_2 -x_1\). In the xy-plane shown, the points \((0, 5)\) and \((1, 9)\) are on line \(j\). It follows that the slope of line \(j\) is \(9 -5/1 -0\), or \(4\). Since the slope of line \(j\) is equal to the slope of line \(k\), the slope of line \(k\) is also \(4\).

The correct answer is \(9\). It's given that the y-intercept of the graph of \(12 x + 2 y = 18\) in the xy-plane is \((0, y)\). Substituting \(0\) for \(x\) in the equation \(12 x + 2 y = 18\) yields \(12(0)+ 2 y = 18\), or \(2 y = 18\). Dividing both sides of this equation by \(2\) yields \(y = 9\). Therefore, the value of \(y\) is \(9\).

Choice D is correct. It’s given that the equation \(80 S + 90 C = 1,120\) represents this situation, where \( S\) is the number of square tokens won, \(C\) is the number of circle tokens won, and \(1,120\) is the total number of points the tokens are worth. It follows that \(80 S\) represents the total number of points the square tokens are worth. Therefore, each square token is worth \(80\) points. It also follows that \(90 C\) represents the total number of points the circle tokens are worth. Therefore, each circle token is worth \(90\) points. Since a circle token is worth \(90\) points and a square token is worth \(80\) points, then a circle token is worth \(90 -80\), or \(10\), more points than a square token. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the number of points a circle token is worth. Choice C is incorrect. This is the number of points a square token is worth.

Choice A is correct. The linear relationship between \(x\) and \(y\) can be represented by the equation \(y = m x + b\), where \(m\) is the slope of the line in the xy-plane that represents the relationship, and \(b\) is the y-coordinate of the y-intercept. The slope can be computed using any two points on the line. The slope of a line between any two points, \((x_1, y_1)\) and \((x_2, y_2)\), on the line can be calculated using the slope formula, \(m = y_2 -y_1/x_2 -x_1\). In the given table, each value of \(x\) and its corresponding value of \(y\) can be represented by a point \((x, y)\). In the given table, when the value of \(x\) is \(1\), the corresponding value of \(y\) is \(11\) and when the value of \(x\) is \(2\), the corresponding value of \(y\) is \(16\). Therefore, the points \((1, 11)\) and \((2, 16)\) are on the line. Substituting \((1, 11)\) and \((2, 16)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the slope formula yields \(m = 16 -11/2 -1\), or \(m = 5\). Substituting \(5\) for \(m\) in the equation \(y = m x + b\) yields \(y = 5 x + b\). Substituting the first value of \(x\) in the table, \(1\), and its corresponding value of \(y\), \(11\), for \(x\) and \(y\), respectively, in this equation yields \(11 = 5(1)+ b\), or \(11 = b + 5\). Subtracting \(5\) from both sides of this equation yields \(6 = b\). Substituting \(6\) for \(b\) in the equation \(y = 5 x + b\) yields \(y = 5 x + 6\). Therefore, the equation \(y = 5 x + 6\) represents the linear relationship between \(x\) and \(y\). Choice B is incorrect. For this relationship, when the value of \(x\) is \(1\), the corresponding value of \(y\) is \(16\), not \(11\). Choice C is incorrect. For this relationship, when the value of \(x\) is \(2\), the corresponding value of \(y\) is \(17\), not \(16\). Choice D is incorrect. For this relationship, when the value of \(x\) is \(1\), the corresponding value of \(y\) is \(17\), not \(11\).

The correct answer is 3. It’s given that Cameron drove 60 miles per hour for x hours; therefore, the distance driven at this speed can be represented by . He then drove 50 miles per hour for y hours; therefore, the distance driven at this speed can be represented by . Since Cameron drove 210 total miles, the equation represents this situation. If, substitution yields, or . Subtracting 60 from both sides of this equation yields . Dividing both sides of this equation by 50 yields .

The correct answer is \(-10\). A line in the xy-plane can be represented by the equation \(y = m x + b\), where \(m\) is the slope of the line and \(b\) is the y-coordinate of the y-intercept. It's given that line \(p\) has a slope of \(-5 thirds\). Therefore, \(m = -5 thirds\). It's also given that line \(p\) has an x-intercept of \((-6, 0)\). Therefore, when \(x = -6\), \(y = 0\). Substituting \(-5 thirds\) for \(m\), \(-6\) for \(x\), and \(0\) for \(y\) in the equation \(y = m x + b\) yields \(0 =(-5 thirds)(-6)+ b\), which is equivalent to \(0 = 10 + b\). Subtracting \(10\) from both sides of this equation yields \(-10 = b\). Therefore, the y-coordinate of the y-intercept of line \(p\) is \(-10\).

Choice D is correct. A linear equation can be written in the form \(y = m x + b\), where \(m\) is the slope of the graph of the equation in the xy-plane and \((0, b)\) is the y-intercept. Subtracting \(10 x\) from each side of the given equation, \(10 x -5 y = -12\), yields \(-5 y = -10 x -12\). Dividing each side of this equation by \(-5\) yields \(y = 2 x + 12/5\). This equation is in the form \(y = m x + b\), where \(m = 2\). Therefore, the slope of the graph of the given equation in the xy-plane is \(2\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice B is correct. It’s given that the employee takes \(1.5\) minutes to prepare a sandwich. Multiplying \(1.5\) by the number of sandwiches, \(x\), yields \(1.5 x\), the amount of time the employee spends preparing \(x\) sandwiches. It’s also given that the employee takes \(1.9\) minutes to prepare a salad. Multiplying \(1.9\) by the number of salads, \(y\), yields \(1.9 y\), the amount of time the employee spends preparing \(y\) salads. It follows that the total amount of time, in minutes, the employee spends preparing \(x\) sandwiches and \(y\) salads is \(1.5 x + 1.9 y\). It's given that the employee spends a total of \(46.1\) minutes preparing \(x\) sandwiches and \(y\) salads. Thus, the equation \(1.5 x + 1.9 y = 46.1\) represents this situation. Choice A is incorrect. This equation represents a situation where it takes the employee \(1.9\) minutes, rather than \(1.5\) minutes, to prepare a sandwich and \(1.5\) minutes, rather than \(1.9\) minutes, to prepare a salad. Choice C is incorrect. This equation represents a situation where it takes the employee \(1\) minute, rather than \(1.5\) minutes, to prepare a sandwich and \(1\) minute, rather than \(1.9\) minutes, to prepare a salad. Choice D is incorrect. This equation represents a situation where it takes the employee \(30.7\) minutes, rather than \(1.5\) minutes, to prepare a sandwich and \(24.3\) minutes, rather than \(1.9\) minutes, to prepare a salad.

Choice D is correct. Each of the tables gives the same three values of \(x\): \(1\), \(2\), and \(4\). Substituting \(1\) for \(x\) in the given equation yields \((3 fifths)(1)+ 3 fourths y = 7\), or \(3 fifths + 3 fourths y = 35/5\). Subtracting \(3 fifths\) from both sides of this equation yields \(3 fourths y = 32/5\). Multiplying both sides of this equation by \(4 thirds\) yields \(y = 128/15\). Therefore, when \(x = 1\), the corresponding value of \(y\) for the given equation is \(128/15\). Substituting \(2\) for \(x\) in the given equation yields \((3 fifths)(2)+ 3 fourths y = 7\), or \(6 fifths + 3 fourths y = 35/5\). Subtracting \(6 fifths\) from both sides of this equation yields \(3 fourths y = 29/5\). Multiplying both sides of this equation by \(4 thirds\) yields \(y = 116/15\). Therefore, when \(x = 2\), the corresponding value of \(y\) for the given equation is \(116/15\). Substituting \(4\) for \(x\) in the given equation yields \((3 fifths)(4)+ 3 fourths y = 7\), or \(12/5 + 3 fourths y = 35/5\). Subtracting \(12/5\) from both sides of this equation yields \(3 fourths y = 23/5\). Multiplying both sides of this equation by \(4 thirds\) yields \(y = 92/15\). Therefore, when \(x = 4\), the corresponding value of \(y\) for the given equation is \(92/15\). The table in choice D gives x-values of \(1\), \(2\), and \(4\) and corresponding y-values of \(128/15\), \(116/15\), and \(92/15\), respectively. Therefore, the table in choice D gives three values of \(x\) and their corresponding values of \(y\) for the given equation. Choice A is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(3 fifths x + 3 fourths + y = 7\). Choice B is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(3 fifths x + y = 10\). Choice C is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(3 fifths x + 3 fourths y = 8\).

Choice A is correct. The slope of the line can be found by choosing any two points on the line, such as (4, –2) and (0, –1). Subtracting the y-values results in –2 – (–1) = –1, the change in y. Subtracting the x-values results in 4 – 0 = 4, the change in x. Dividing the change in y by the change in x yields, the slope. The line intersects the y-axis at (0, –1), so –1 is the y-coordinate of the y-intercept. This information can be expressed in slope-intercept form as the equation .Choice B is incorrect and may result from incorrectly calculating the slope and then misidentifying the slope as the y-intercept. Choice C is incorrect and may result from misidentifying the slope as the y-intercept. Choice D is incorrect and may result from incorrectly calculating the slope.

The correct answer is 74. The y-intercept of a line in the xy-plane is the ordered pair of the point of intersection of the line with the y-axis. Since line k intersects the y-axis at the point, it follows that is the y-intercept of this line. An equation of any line in the xy-plane can be written in the form, where m is the slope of the line and b is the y-coordinate of the y-intercept. Therefore, the equation of line k can be written as, or . The value of m can be found by substituting the x-and y-coordinates from a point on the line, such as, for x and y, respectively. This results in . Solving this equation for m gives . Therefore, an equation of line k is . The value of w can be found by substituting the x-coordinate, 20, for x in the equation of line k and solving this equation for y. This gives, or . Since w is the y-coordinate of this point, .

Choice B is correct. It’s given that the equation models the relationship between the number of differentπeces of music a certainπanist practices, y, and the number of minutes in a practice session, x. Since it’s given that the session lasted 30 minutes, the number ofπeces theπanist practiced can be found by substituting 30 for x in the given equation, which yields, or .Choices A and C are incorrect and may result from misinterpreting the values in the equation. Choice D is incorrect. This is the given value of x, not the value of y.

Choice A is correct. When the graph of an equation in the form \(A x + B y = C\), where \(A\), \(B\), and \(C\) are constants, is shifted down \(k\) units in the xy-plane, the resulting graph can be represented by the equation \(A x + B(y + k)= C\). It's given that the graph of \(27 x + 33 y = 297\) is shifted down \(5\) units in the xy-plane. Therefore, the resulting graph can be represented by the equation \(27 x + 33(y + 5)= 297\), or \(27 x + 33 y + 165 = 297\). Subtracting \(165\) from both sides of this equation yields \(27 x + 33 y = 132\). The y-intercept of the graph of an equation in the xy-plane is the point where the line intersects the y-axis, represented by the point \((0, y)\). Substituting \(0\) for \(x\) in the equation \(27 x + 33 y = 132\) yields \(27(0)+ 33 y = 132\), or \(33 y = 132\). Dividing both sides of this equation by \(33\) yields \(y = 4\). Therefore, if the graph of \(27 x + 33 y = 297\) is shifted down \(5\) units, the y-intercept of the resulting graph is \((0, 4)\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the y-intercept of the graph of \(27 x + 33 y = 297\) shifted up, not down, \(5\) units. Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. It's given that \(364\) paper straws of equal length were used to construct triangles and rectangles, where no two polygons had a common side. It's also given that the equation \(3 x + 4 y = 364\) represents this situation, where \(x\) is the number of triangles constructed and \(y\) is the number of rectangles constructed. The equation \((x, y)=(24, 73)\) means that if \(x = 24\), then \(y = 73\). Substituting \(24\) for \(x\) and \(73\) for \(y\) in \(3 x + 4 y = 364\) yields \(3(24)+ 4(73)= 364\), or \(364 = 364\), which is true. Therefore, in this context, the equation \((x, y)=(24, 73)\) means that if \(24\) triangles were constructed, then \(73\) rectangles were constructed. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice D is correct. A line in the xy-plane with a slope of \(m\) and a y-intercept of \((0, b)\) can be defined by an equation in the form \(y = m x + b\). It’s given that line \(r\) has a slope of \(4\) and passes through the point \((0, 6)\). It follows that \(m = 4\) and \(b = 6\). Substituting \(4\) for \(m\) and \(6\) for \(b\) in the equation \(y = m x + b\) yields \(y = 4 x + 6\). Therefore, the equation \(y = 4 x + 6\) defines line \(r\). Choice A is incorrect. This equation defines a line that has a slope of \(-6\), not \(4\), and passes through the point \((0, 4)\), not \((0, 6)\). Choice B is incorrect. This equation defines a line that has a slope of \(6\), not \(4\), and passes through the point \((0, 4)\), not \((0, 6)\). Choice C is incorrect. This equation defines a line that passes through the point \((0, -6)\), not \((0, 6)\).

Choice B is correct. It’s given that it takes the machine \(10\) minutes to make a large box. It's also given that \(x\) represents the possible number of large boxes the machine can make each day. Multiplying \(10\) by \(x\) gives \(10 x\), which represents the amount of time spent making large boxes. It’s given that it takes the machine \(5\) minutes to make a small box. It's also given that \(y\) represents the possible number of small boxes the machine can make each day. Multiplying \(5\) by \(y\) gives \(5 y\), which represents the amount of time spent making small boxes. Combining the amount of time spent making \(x\) large boxes and \(y\) small boxes yields \(10 x + 5 y\). It’s given that the machine makes boxes for a total of \(700\) minutes each day. Therefore \(10 x + 5 y = 700\) represents the possible number of large boxes, \(x\), and small boxes, \(y\), the machine can make each day. Choice A is incorrect and may result from associating the time of \(10\) minutes with small, rather than large, boxes and the time of \(5\) minutes with large, rather than small, boxes. Choice C is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice D is correct. An equation defining a line in the xy-plane can be written in the form \(y = m x + b\), where \(m\) represents the slope and \((0, b)\) represents the y-intercept of the line. It’s given that line \(t\) passes through the point \((0, 9)\); therefore, \(b = 9\). The slope, \(m\), of a line can be found using any two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), and the slope formula \(m = y_2 -y_1/x_2 -x_1\). Substituting \((0, 9)\) and \((1, 17)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the slope formula yields \(m = 17 -9/1 -0\), or \(m = 8\). Substituting \(8\) for \(m\) and \(9\) for \(b\) in the equation \(y = m x + b\) yields \(y = 8 x + 9\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The equation that defines line \(t\) in the xy-plane can be written in slope-intercept form \(y = m x + b\), where \(m\) is the slope of line \(t\) and \((0, b)\) is its y-intercept. It’s given that line \(t\) has a slope of \(-1 third\). Therefore, \(m = -1 third\). Substituting \(-1 third\) for \(m\) in the equation \(y = m x + b\) yields \(y = -1 third x + b\), or \(y = -x/3 + b\). It’s also given that line \(t\) passes through the point \((9, 10)\). Substituting \(9\) for \(x\) and \(10\) for \(y\) in the equation \(y = -x/3 + b\) yields \(10 = -9 thirds + b\), or \(10 = -3 + b\). Adding \(3\) to both sides of this equation yields \(13 = b\). Substituting \(13\) for \(b\) in the equation \(y = -x/3 + b\) yields \(y = -x/3 + 13\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This equation defines a line that has a slope of \(9\), not \(-1 third\), and passes through the point \((0, 10)\), not \((9, 10)\). Choice C is incorrect. This equation defines a line that passes through the point \((0, 10)\), not \((9, 10)\).

The correct answer is \(46\). It's given that a chemist combines water and acetic acid to make a mixture with a volume of \(56\) milliliters (mL) and that the volume of acetic acid in the mixture is \(10\) mL. Let \(x\) represent the volume of water, in mL, in the mixture. The equation \(x + 10 = 56\) represents this situation. Subtracting \(10\) from both sides of this equation yields \(x = 46\). Therefore, the volume of water, in mL, in the mixture is \(46\).

Choice A is correct. It's given that a certain township consists of a \(5\)-hectare industrial park and a \(24\)-hectare neighborhood and that the total number of trees in the township is \(4,529\). It's also given that the equation \(5 x + 24 y = 4,529\) represents this situation. Since the total number of trees for a given area can be determined by taking the size of the area, in hectares, times the average number of trees per hectare, the best interpretation of \(5 x\) is the number of trees in the industrial park and the best interpretation of \(24 y\) is the number of trees in the neighborhood. Since \(5\) is the size of the industrial park, in hectares, the best interpretation of \(x\) is the average number of trees per hectare in the industrial park. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice C is correct. Maria spends x minutes running each day and y minutes biking each day. Therefore, represents the total number of minutes Maria spent running and biking each day. Because, it follows that 75 is the total number of minutes that Maria spent running and biking each day.Choices A and B are incorrect. The number of minutes Maria spent running each day is represented by x and need not be 75. Similarly, the number of minutes that Maria spends biking each day is represented by y and need not be 75. The number of minutes Maria spends running each day and biking each day may vary; however, the total number of minutes she spends each day on these activities is constant and equal to 75. Choice D is incorrect. The number of minutes Maria spent biking for each minute spent running cannot be determined from the information provided.

Choice B is correct. The equation of a line in the xy-plane can be written in slope-intercept form \(y = m x + b\), where \(m\) is the slope of the line and \((0, b)\) is its y-intercept. It’s given that the line passes through the point \((0, 5)\). Therefore, \(b = 5\). It’s also given that the line is parallel to the graph of \(y = 7 x + 4\), which means the line has the same slope as the graph of \(y = 7 x + 4\). The slope of the graph of \(y = 7 x + 4\) is \(7\). Therefore, \(m = 7\). Substituting \(7\) for \(m\) and \(5\) for \(b\) in the equation \(y = m x + b\) yields \(y = 7 x + 5\). Choice A is incorrect. The graph of this equation passes through the point \((0, 0)\), not \((0, 5)\), and has a slope of \(5\), not \(7\). Choice C is incorrect. The graph of this equation passes through the point \((0, 0)\), not \((0, 5)\). Choice D is incorrect. The graph of this equation passes through the point \((0, 7)\), not \((0, 5)\), and has a slope of \(5\), not \(7\).

Choice B is correct. It’s given that a represents the number of small boxes and b represents the number of large boxes the customer had shipped. If the customer had 3 small boxes shipped, then . Substituting 3 for a in the equation yields or . Subtracting 9 from both sides of the equation yields . Dividing both sides of this equation by 4 yields . Therefore, the customer had 4 large boxes shipped.Choices A, C, and D are incorrect. If the number of large boxes shipped is 3, then . Substituting 3 for b in the given equation yields or . Subtracting 12 from both sides of the equation and then dividing by 3 yields . However, it’s given that the number of small boxes shipped, a, is 3, not, so b cannot equal 3. Similarly, if or, then or, respectively, which is also not true.

Choice A is correct. Substituting \(0\) for \(x\) into the given equation yields \(y = 0 + 4\), or \(y = 4\). Therefore, when \(x = 0\), the corresponding value of \(y\) for the given equation is \(4\). Substituting \(1\) for \(x\) into the given equation yields \(y = 1 + 4\), or \(y = 5\). Therefore, when \(x = 1\), the corresponding value of \(y\) for the given equation is \(5\). Substituting \(2\) for \(x\) into the given equation yields \(y = 2 + 4\), or \(y = 6\). Therefore, when \(x = 2\), the corresponding value of \(y\) for the given equation is \(6\). Of the choices given, only the table in choice A gives these three values of \(x\) and their corresponding values of \(y\) for the given equation. Choice B is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(y = -x + 6\). Choice C is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(y = -x + 2\). Choice D is incorrect. This table gives three values of \(x\) and their corresponding values of \(y\) for the equation \(y = x\).

Choice B is correct. It's given that line \(k\) is defined by \(y = 17 x/7 + 4\). For an equation of a line written in the form \(y = m x + b\), \(m\) is the slope of the line and \(b\) is the y-coordinate of the y-intercept of the line. It follows that the slope of line \(k\) is \(17/7\). It's also given that line \(j\) is parallel to line \(k\) in the xy-plane. Since parallel lines have equal slopes, line \(j\) also has a slope of \(17/7\). Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the y-coordinate of the y-intercept of line \(k\), not the slope of line \(j\). Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. Let \(d\) represent the mass, in grams, of vitamin D in the mixture, and let \(c\) represent the mass, in grams, of calcium in the mixture. It’s given that the mixture consists of only vitamin D and calcium and that the total mass of the mixture is \(150\) grams. Therefore, the equation \(d + c = 150\) represents this situation. It’s also given that the mass of vitamin D in the mixture is \(50\) grams. Substituting \(50\) for \(d\) in the equation \(d + c = 150\) yields \(50 + c = 150\). Subtracting \(50\) from both sides of this equation yields \(c = 100\). Therefore, the mass of calcium in the mixture is \(100\) grams. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the total mass, in grams, of the mixture, not the mass, in grams, of calcium in the mixture. Choice D is incorrect. This is the mass, in grams, of vitamin D in the mixture, not the mass, in grams, of calcium in the mixture.

Choice C is correct. It’s given that line \(r\) is perpendicular to line \(p\) in the xy-plane. This means that the slope of line \(r\) is the -reciprocal of the slope of line \(p\). If the equation for line \(p\) is rewritten in slope-intercept form \(y = m x + b\), where \(m\) and \(b\) are constants, then \(m\) is the slope of the line and \((0, b)\) is its y-intercept. Subtracting \(18 x\) from both sides of the equation \(2 y + 18 x = 9\) yields \(2 y = -18 x + 9\). Dividing both sides of this equation by \(2\) yields \(y = -9 x + 9 halves\). It follows that the slope of line \(p\) is \(-9\). The -reciprocal of a number is \(-1\) divided by the number. Therefore, the -reciprocal of \(-9\) is \( -1/ -9\), or \(1 ninth\). Thus, the slope of line \(r\) is \(1 ninth\). Choice A is incorrect. This is the slope of line \(p\), not line \(r\). Choice B is incorrect. This is the reciprocal, not the -reciprocal, of the slope of line \(p\). Choice D is incorrect. This is the -, not the -reciprocal, of the slope of line \(p\).

The correct answer is \(1 half\). For an equation in slope-intercept form \(y = m x + b\), \(m\) represents the slope of the line in the xy-plane defined by this equation. It's given that line \(p\) is defined by \(4 y + 8 x = 6\). Subtracting \(8 x\) from both sides of this equation yields \(4 y = -8 x + 6\). Dividing both sides of this equation by \(4\) yields \(y = -8 fourths x + 6 fourths\), or \(y = -2 x + 3 halves\). Thus, the slope of line \(p\) is \(-2\). If line \(r\) is perpendicular to line \(p\), then the slope of line \(r\) is the -reciprocal of the slope of line \(p\). The -reciprocal of \(-2\) is \(-1/(-2) = 1 half\). Note that 1/2 and .5 are examples of ways to enter a correct answer.

Choice A is correct. It's given that line \(j\) is perpendicular to line \(k\) in the xy-plane. It follows that the slope of line \(j\) is the opposite reciprocal of the slope of line \(k\). The equation for line \(k\) is written in slope-intercept form \(y = m x + b\), where \(m\) is the slope of the line and \(b\) is the y-coordinate of the y-intercept of the line. It follows that the slope of line \(k\) is \(3\). The opposite reciprocal of a number is \(-1\) divided by the number. Thus, the opposite reciprocal of \(3\) is \(-1 third\). Therefore, the slope of line \(j\) is \(-1 third\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. It's given that the store's sales of a certain Greek yogurt totaled \(1,277.94\) dollars last month. It's also given that the equation \(5.48 x + 7.30 y = 1,277.94\) represents this situation, where \(x\) is the number of smaller containers sold and \(y\) is the number of larger containers sold. Since \(x\) represents the number of smaller containers of yogurt sold, the expression \(5.48 x\) represents the total sales, in dollars, from smaller containers of yogurt. This means that \(x\) smaller containers of yogurt were sold at a price of \(5.48\) dollars each. Therefore, according to the equation, \(5.48\) represents the price, in dollars, of each smaller container. Choice B is incorrect. This expression represents the total sales, in dollars, from selling \(y\) larger containers of yogurt. Choice C is incorrect. This value represents the price, in dollars, of each larger container of yogurt. Choice D is incorrect. This expression represents the total sales, in dollars, from selling \(x\) smaller containers of yogurt.

The correct answer is \(5\). It's given that the equation \(10 x + 15 y = 85\) represents the situation, where \(x\) is the number of on-site training courses, \(y\) is the number of online training courses, and \(85\) is the total number of hours of training courses the apprentice has enrolled in. Therefore, \(10 x\) represents the number of hours the apprentice has enrolled in on-site training courses, and \(15 y\) represents the number of hours the apprentice has enrolled in online training courses. Since \(x\) is the number of on-site training courses and \(y\) is the number of online training courses the apprentice has enrolled in, \(10\) is the number of hours each on-site course takes and \(15\) is the number of hours each online course takes. Subtracting these numbers gives \(15 -10\), or \(5\) more hours each online training course takes than each on-site training course.

Choice D is correct. The equation of line \(h\) can be written in slope-intercept form \(y = m x + b\), where \(m\) is the slope of the line and \((0, b)\) is the y-intercept of the line. It’s given that line \(h\) contains the points \((18, 130)\), \((23, 160)\), and \((26, 178)\). Therefore, its slope \(m\) can be found as \(160 -130/23 -18\), or \(6\). Substituting \(6\) for \(m\) in the equation \(y = m x + b\) yields \(y = 6 x + b\). Substituting \(130\) for \(y\) and \(18\) for \(x\) in this equation yields \(130 = 6(18)+ b\), or \(130 = 108 + b\). Subtracting \(108\) from both sides of this equation yields \(22 = b\). Substituting \(22\) for \(b\) in \(y = 6 x + b\) yields \(y = 6 x + 22\). Since line \(k\) is the result of translating line \(h\) down \(5\) units, an equation of line \(k\) is \(y = 6 x + 22 -5\), or \(y = 6 x + 17\). Substituting \(0\) for \(y\) in this equation yields \(0 = 6 x + 17\). Solving this equation for \(x\) yields \(x = -17/6\). Therefore, the x-intercept of line \(k\) is \((-17/6, 0)\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

The correct answer is \(1 fourth\). For an equation in slope-intercept form \(y = m x + b\), \(m\) represents the slope of the line in the xy-plane defined by this equation. It's given that line \(ell\) is defined by \(3 y + 12 x = 5\). Subtracting \(12 x\) from both sides of this equation yields \(3 y = -12 x + 5\). Dividing both sides of this equation by \(3\) yields \(y = -12/3 x + 5 thirds\), or \(y = -4 x + 5 thirds\). Thus, the slope of line \(ell\) in the xy-plane is \(-4\). Since line \(n\) is perpendicular to line \(ell\) in the xy-plane, the slope of line \(n\) is the -reciprocal of the slope of line \(ell\). The -reciprocal of \(-4\) is \(-1/(-4) = 1 fourth\). Note that 1/4 and .25 are examples of ways to enter a correct answer.

Choice A is correct. The value of k can be found using the slope-intercept form of a linear equation,, where m is the slope and b is the y-coordinate of the y-intercept. The equation can be rewritten in the form . One of the given points,, is the y-intercept. Thus, the y-coordinate of the y-intercept must be equal to . Multiplying both sides by k gives . Dividing both sides by gives .Choices B, C, and D are incorrect and may result from errors made rewriting the given equation.

The correct answer is \(33\). It’s given in the table that the coordinates of two points on a line in the xy-plane are \((k, 13)\) and \((k + 7, -15)\). The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, \(m\), between any two points, \((x_1, y_1)\) and \((x_2, y_2)\), on the line can be calculated using the slope formula, \(m = (y_2 -y_1)/(x_2 -x_1)\). It follows that the slope of the line with the given points from the table, \((k, 13)\) and \((k + 7, -15)\), is \(m = -15 -13/k + 7 -k\), which is equivalent to \(m = -28/7\), or \(m = -4\). It's given that the y-intercept of the line is \((k -5, b)\). Substituting \(-4\) for \(m\) and the coordinates of the points \((k -5, b)\) and \((k, 13)\) into the slope formula yields \(-4 = 13 -b/k -(k -5)\), which is equivalent to \(-4 = 13 -b/k -k + 5\), or \(-4 = 13 -b/5\). Multiplying both sides of this equation by \(5\) yields \(-20 = 13 -b\). Subtracting \(13\) from both sides of this equation yields \(-33 = -b\). Dividing both sides of this equation by \(-1\) yields \(b = 33\). Therefore, the value of \(b\) is \(33\).

The correct answer is \(3/17\). It’s given that line \(j\) is perpendicular to line \(k\) in the xy-plane. This means that the slope of line \(j\) is the -reciprocal of the slope of line \(k\). The equation of line \(k\), \(y = -17/3 x + 5\), is written in slope-intercept form \(y = m x + b\), where \(m\) is the slope of the line and \(b\) is the y-coordinate of the y-intercept of the line. It follows that the slope of line \(k\) is \(-17/3\). The -reciprocal of a number is \(-1\) divided by the number. Therefore, the -reciprocal of \(-17/3\) is \( -1/ -17/3\), or \(3 seventeenths\). Thus, the slope of line \(j\) is \(3 seventeenths\). Note that 3/17, .1764, .1765, and 0.176 are examples of ways to enter a correct answer.