To find the equations of a line parallel and perpendicular to the given line, we first need to find the slope of the line represented by the given table.
The table provides the following points:
- \((3, -8)\)
- \((4, -10)\)
- \((5, -12)\)
- \((6, -14)\)
To determine the slope \(m\), we can use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((3, -8)\) and \((4, -10)\): \[ m = \frac{-10 - (-8)}{4 - 3} = \frac{-10 + 8}{1} = \frac{-2}{1} = -2 \]
So, the slope of the line represented by the table is \(m = -2\).
a. Equation of a Line Parallel to the Given Line
A line parallel to the given line will have the same slope, which is \(-2\). We need to use this slope and the indicated y-intercept to write the equation.
Let's say the indicated y-intercept is \(b\). The equation of the line in slope-intercept form \(y = mx + b\) will be:
\[ y = -2x + b \]
(Please provide the indicated y-intercept if you want a specific equation.)
b. Equation of a Line Perpendicular to the Given Line
The slope of a line that is perpendicular to another is the negative reciprocal of the slope of the original line.
Given the slope of the original line is \(-2\), the slope of the perpendicular line will be: \[ m_{\text{perpendicular}} = -\frac{1}{-2} = \frac{1}{2} \]
Again, let's assume the indicated y-intercept for the perpendicular line is \(b'\). Then the equation of the line will be:
\[ y = \frac{1}{2}x + b' \]
(Please provide the indicated y-intercept if you want a specific equation.)
In summary, we need the y-intercepts to provide specific equations. If you provide \(b\) and \(b'\), I can finalize those equations for you.