Linear Equations - 12 Questions with Answers + Explanations

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Substituting \(m = -4\) and \(b = 5\), the equation is \(y = -4x + 5\).

To find the slope of the line, rearrange the equation into slope-intercept form \(y = mx + b\), where \(m\) is the slope. Start with \(2x + 3y = 12\): \(3y = -2x + 12\), then divide through by 3 to get \(y = -\frac{2}{3}x + 4\). The slope is \(-\frac{2}{3}\).

The slopes of both lines are equal (\(m = 2\)), indicating the lines are parallel. Parallel lines never intersect.

The x-intercept occurs when \(y = 0\). Substitute \(y = 0\) into the equation: \(4x - 6(0) = 24\), so \(4x = 24\). Divide by 4 to get \(x = 6\). Therefore, the x-intercept is \(6\).

To find the equation of the line, start by calculating the slope, \(m\), using the formula:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Substitute the points \((2, 5)\) and \((4, 11)\):

\(m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3\)

Now, use the point-slope form \(y - y_1 = m(x - x_1)\) with the point \((2, 5)\):

\(y - 5 = 3(x - 2)\)

Simplify to slope-intercept form:

\(y = 3x - 6 + 5\)

Final equation: \(y = 3x - 1\).

First, find the slope of the given line by rearranging it into slope-intercept form, \(y = mx + b\). Start with:

\(3x - 2y = 6\)

Solve for \(y\):

\(-2y = -3x + 6\)

\(y = \frac{3}{2}x - 3\)

The slope of the given line is \(\frac{3}{2}\). The slope of a line perpendicular to this is the negative reciprocal of \(\frac{3}{2}\):

\(m = -\frac{2}{3}\).

The y-intercept of a line is the point where it intersects the y-axis. This occurs when \(x = 0\). Substitute \(x = 0\) into the equation:

\(y = -2(0) + 7\)

\(y = 7\)

The y-coordinate of the intersection point is 7, and the point is \((0, 7)\).

The point-slope form of a line is:

\(y - y_1 = m(x - x_1)\)

Substitute the given slope \(m = 5\) and point \((x_1, y_1) = (3, -2)\):

\(y - (-2) = 5(x - 3)\)

Simplify the double negative:

\(y + 2 = 5(x - 3)\)

The equation of the line in point-slope form is \(y + 2 = 5(x - 3)\).

The x-intercept of a line is the point where it intersects the x-axis. This occurs when \(y = 0\). Substitute \(y = 0\) into the equation:

\(0 = 4x - 8\)

Solve for \(x\):

\(4x = 8\)

\(x = 2\)

The x-coordinate of the intersection point is \(2\), and the point is \((2, 0)\).

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Substitute \((x_1, y_1) = (1, -3)\) and \((x_2, y_2) = (5, 9)\):

\(m = \frac{9 - (-3)}{5 - 1} = \frac{9 + 3}{4} = \frac{12}{4} = 3\)

The slope of the line is \(3\).

Two lines are parallel if they have the same slope. The given line is \(y = \frac{1}{2}x + 4\), so its slope is \(\frac{1}{2}\). Any equation with the same slope but a different y-intercept will be parallel. For example:

\(y = \frac{1}{2}x - 3\) has the same slope (\(\frac{1}{2}\)) but a different y-intercept.

Thus, the correct answer is \(y = \frac{1}{2}x - 3\).

The y-intercept of a line occurs when \(x = 0\). Substitute \(x = 0\) into the equation:

\(2(0) + 3y = 12\)

\(3y = 12\)

\(y = \frac{12}{3} = 4\)

The y-intercept is \((0, 4)\), so the y-coordinate of the intercept is \(4\).