Question

consider the line $y = \frac{4}{3}x - 7$. find the equation of the line that is parallel to this line and passes through the point $(8, 2)$. find the equation of the line that is perpendicular to this line and passes through the point $(8, 2)$. note that the aleks graphing calculator may be helpful in checking your answer. equation of parallel line: equation of perpendicular line:

Explanation:

Step1: Recall slope of parallel lines

Parallel lines have the same slope. The given line is \( y = \frac{4}{3}x - 7 \), so the slope \( m_{\text{parallel}} = \frac{4}{3} \).

Step2: Use point - slope form for parallel line

Point - slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(8,2) \) and \( m = \frac{4}{3} \).
Substitute values: \( y - 2=\frac{4}{3}(x - 8) \)
Expand: \( y - 2=\frac{4}{3}x-\frac{32}{3} \)
Add 2 to both sides: \( y=\frac{4}{3}x-\frac{32}{3}+\frac{6}{3}=\frac{4}{3}x-\frac{26}{3} \)

Step3: Recall slope of perpendicular lines

Perpendicular lines have slopes that are negative reciprocals. So \( m_{\text{perpendicular}}=-\frac{3}{4} \) (since the reciprocal of \( \frac{4}{3} \) is \( \frac{3}{4} \) and we change the sign).

Step4: Use point - slope form for perpendicular line

Using point - slope form with \( (x_1,y_1)=(8,2) \) and \( m = -\frac{3}{4} \):
\( y - 2=-\frac{3}{4}(x - 8) \)
Expand: \( y - 2=-\frac{3}{4}x + 6 \)
Add 2 to both sides: \( y=-\frac{3}{4}x+8 \)

Answer:

Equation of parallel line: \( y=\frac{4}{3}x-\frac{26}{3} \)
Equation of perpendicular line: \( y = -\frac{3}{4}x + 8 \)