Question

consider the line $y = -9x + 8$. find the equation of the line that is perpendicular to this line and passes through the point $(-5, 4)$. find the equation of the line that is parallel to this line and passes through the point $(-5, 4)$. note that the aleks graphing calculator may be helpful in checking your answer. equation of perpendicular line: equation of parallel line:

Explanation:

Step1: Recall slope rules

For a line \( y = mx + b \), slope is \( m \). Parallel lines have equal slopes (\( m_1 = m_2 \)). Perpendicular lines have slopes that are negative reciprocals (\( m_1 \times m_2 = -1 \)).

Given line: \( y = -9x + 8 \), so slope \( m = -9 \).

Step2: Perpendicular line slope

Let \( m_p \) be slope of perpendicular line. Then \( -9 \times m_p = -1 \) → \( m_p = \frac{1}{9} \).

Use point-slope form \( y - y_1 = m(x - x_1) \), point \( (-5, 4) \):

\( y - 4 = \frac{1}{9}(x - (-5)) \) → \( y - 4 = \frac{1}{9}(x + 5) \)

Simplify: \( y = \frac{1}{9}x + \frac{5}{9} + 4 \) → \( y = \frac{1}{9}x + \frac{5}{9} + \frac{36}{9} \) → \( y = \frac{1}{9}x + \frac{41}{9} \)

Step3: Parallel line slope

Slope of parallel line \( m_{\parallel} = -9 \) (same as original line).

Use point-slope form with point \( (-5, 4) \):

\( y - 4 = -9(x - (-5)) \) → \( y - 4 = -9(x + 5) \)

Simplify: \( y - 4 = -9x - 45 \) → \( y = -9x - 45 + 4 \) → \( y = -9x - 41 \)

Answer:

Equation of perpendicular line: \( y = \frac{1}{9}x + \frac{41}{9} \)
Equation of parallel line: \( y = -9x - 41 \)