Question

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

Explanation:

Step1: Recall slope of parallel lines

Parallel lines have the same slope. The given line is \( y = -\frac{3}{5}x + 9 \), so its slope \( m = -\frac{3}{5} \). The parallel line through \( (5, 2) \) has the same slope.
Using point - slope form \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(5,2) \) and \( m = -\frac{3}{5} \):
\( y - 2=-\frac{3}{5}(x - 5) \)
Simplify: \( y - 2=-\frac{3}{5}x+3 \), then \( y = -\frac{3}{5}x + 5 \)

Step2: Recall slope of perpendicular lines

The slope of a line perpendicular to a line with slope \( m \) is \( m_{\perp}=-\frac{1}{m} \). For \( m = -\frac{3}{5} \), \( m_{\perp}=\frac{5}{3} \)
Using point - slope form with \( (x_1,y_1)=(5,2) \) and \( m=\frac{5}{3} \):
\( y - 2=\frac{5}{3}(x - 5) \)
Simplify: \( y - 2=\frac{5}{3}x-\frac{25}{3} \), then \( y=\frac{5}{3}x-\frac{25}{3}+2=\frac{5}{3}x-\frac{25 - 6}{3}=\frac{5}{3}x-\frac{19}{3} \)

Answer:

Equation of parallel line: \( y = -\frac{3}{5}x + 5 \)
Equation of perpendicular line: \( y=\frac{5}{3}x-\frac{19}{3} \)