Question

consider the line $y = \frac{5}{3}x - 2$. find the equation of the line that is perpendicular to this line and passes through the point $(9, -3)$. find the equation of the line that is parallel to this line and passes through the point $(9, -3)$. 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 properties

For a line \( y = mx + b \), slope is \( m \). Given line: \( y=\frac{5}{3}x - 2 \), so slope \( m_1=\frac{5}{3} \).

Perpendicular Line:

Step2: Find perpendicular slope

Perpendicular slope \( m_2 = -\frac{1}{m_1}=-\frac{3}{5} \).

Step3: Use point - slope form

Point - slope: \( y - y_1 = m(x - x_1) \), point \( (9,-3) \), \( m = -\frac{3}{5} \).
\( y - (-3)=-\frac{3}{5}(x - 9) \)
\( y + 3=-\frac{3}{5}x+\frac{27}{5} \)
\( y=-\frac{3}{5}x+\frac{27}{5}-3 \)
\( y=-\frac{3}{5}x+\frac{27 - 15}{5}=-\frac{3}{5}x+\frac{12}{5} \)

Parallel Line:

Step4: Find parallel slope

Parallel slope \( m_3=m_1=\frac{5}{3} \).

Step5: Use point - slope form

Point \( (9,-3) \), \( m=\frac{5}{3} \).
\( y - (-3)=\frac{5}{3}(x - 9) \)
\( y + 3=\frac{5}{3}x-15 \)
\( y=\frac{5}{3}x-15 - 3 \)
\( y=\frac{5}{3}x-18 \)

Answer:

Equation of perpendicular line: \( y = -\frac{3}{5}x+\frac{12}{5} \)
Equation of parallel line: \( y=\frac{5}{3}x - 18 \)