Question

consider the line $y = -\frac{8}{5}x + 9$. find the equation of the line that is parallel to this line and passes through the point $(-8, 6)$. find the equation of the line that is perpendicular to this line and passes through the point $(-8, 6)$. note that the aleks graphing calculator may be helpful in checking your answer. equation of parallel line: equation of perpendicular line:

Explanation:

Step1: Identify parallel slope

Parallel lines have equal slopes. The given line is $y = -\frac{8}{5}x + 9$, so slope $m_{\text{parallel}} = -\frac{8}{5}$.

Step2: Find parallel line equation

Use point-slope form $y - y_1 = m(x - x_1)$ with $(x_1, y_1)=(-8,6)$:
$y - 6 = -\frac{8}{5}(x - (-8))$
Simplify:
$y - 6 = -\frac{8}{5}(x + 8)$
$y - 6 = -\frac{8}{5}x - \frac{64}{5}$
$y = -\frac{8}{5}x - \frac{64}{5} + \frac{30}{5}$
$y = -\frac{8}{5}x - \frac{34}{5}$

Step3: Identify perpendicular slope

Perpendicular slopes are negative reciprocals: $m_{\text{perpendicular}} = \frac{5}{8}$.

Step4: Find perpendicular line equation

Use point-slope form with $(x_1, y_1)=(-8,6)$:
$y - 6 = \frac{5}{8}(x - (-8))$
Simplify:
$y - 6 = \frac{5}{8}(x + 8)$
$y - 6 = \frac{5}{8}x + 5$
$y = \frac{5}{8}x + 5 + 6$
$y = \frac{5}{8}x + 11$

Answer:

Equation of parallel line: $y = -\frac{8}{5}x - \frac{34}{5}$
Equation of perpendicular line: $y = \frac{5}{8}x + 11$