Questions: Consider the line y=-5x+7. Find the equation of the line that is parallel to this line and passes through the point (-7,5). Find the equation of the line that is perpendicular to this line and passes through the point (-7,5).

Consider the line y=-5x+7.
Find the equation of the line that is parallel to this line and passes through the point (-7,5).
Find the equation of the line that is perpendicular to this line and passes through the point (-7,5).

Transcript text: Consider the line $y=-5 x+7$. Find the equation of the line that is parallel to this line and passes through the point $(-7,5)$. Find the equation of the line that is perpendicular to this line and passes through the point $(-7,5)$.

Solution

Solution Steps

Step 1: Identify the slope of the given line

The given line is $ y = -5x + 7 $. The slope of this line is $ m = -5 $.

Step 2: Find the equation of the parallel line

A line parallel to the given line will have the same slope, $ m = -5 $. Using the point-slope form of a line, $ y - y_1 = m(x - x_1) $, and the point $(-7, 5)$, we substitute the values: \[ y - 5 = -5(x - (-7)) \] Simplify the equation: \[ y - 5 = -5(x + 7) \] \[ y - 5 = -5x - 35 \] \[ y = -5x - 30 \]

Step 3: Find the slope of the perpendicular line

A line perpendicular to the given line will have a slope that is the negative reciprocal of $ m = -5 $. Thus, the slope of the perpendicular line is: \[ m_{\text{perpendicular}} = \frac{1}{5} \]

Step 4: Find the equation of the perpendicular line

Using the point-slope form again with the slope $ m = \frac{1}{5} $ and the point $(-7, 5)$, we substitute the values: \[ y - 5 = \frac{1}{5}(x - (-7)) \] Simplify the equation: \[ y - 5 = \frac{1}{5}(x + 7) \] \[ y - 5 = \frac{1}{5}x + \frac{7}{5} \] \[ y = \frac{1}{5}x + \frac{7}{5} + 5 \] \[ y = \frac{1}{5}x + \frac{7}{5} + \frac{25}{5} \] \[ y = \frac{1}{5}x + \frac{32}{5} \]