Question

consider the line $y = -8x + 6$. find the equation of the line that is perpendicular to this line and passes through the point $(-2, -6)$. find the equation of the line that is parallel to this line and passes through the point $(-2, -6)$. 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: \( y = -8x + 6 \), so slope \( m = -8 \).

Step2: Find perpendicular slope

Let \( m_p \) be perpendicular slope. \( -8 \times m_p = -1 \Rightarrow m_p = \frac{1}{8} \).

Step3: Perpendicular line equation

Use point-slope form \( y - y_1 = m(x - x_1) \), point \( (-2, -6) \), \( m = \frac{1}{8} \):
\( y - (-6) = \frac{1}{8}(x - (-2)) \Rightarrow y + 6 = \frac{1}{8}(x + 2) \Rightarrow y = \frac{1}{8}x + \frac{2}{8} - 6 \Rightarrow y = \frac{1}{8}x + \frac{1}{4} - \frac{24}{4} \Rightarrow y = \frac{1}{8}x - \frac{23}{4} \).

Step4: Find parallel slope

Parallel slope \( m_{||} = -8 \) (same as original line).

Step5: Parallel line equation

Use point-slope form, \( m = -8 \), point \( (-2, -6) \):
\( y - (-6) = -8(x - (-2)) \Rightarrow y + 6 = -8(x + 2) \Rightarrow y + 6 = -8x - 16 \Rightarrow y = -8x - 22 \).

Answer:

Equation of perpendicular line: \( y = \frac{1}{8}x - \frac{23}{4} \)
Equation of parallel line: \( y = -8x - 22 \)