Question

consider the line $y = \frac{3}{2}x - 4$. find the equation of the line that is parallel to this line and passes through the point $(9, 3)$. find the equation of the line that is perpendicular 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 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 \( y = \frac{3}{2}x - 4 \), so the slope \( m_{\text{parallel}}=\frac{3}{2} \).

Step2: Use point - slope form for parallel line

Point - slope form is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(9,3) \) and \( m = \frac{3}{2} \).
Substitute values: \( y - 3=\frac{3}{2}(x - 9) \)
Expand: \( y-3=\frac{3}{2}x-\frac{27}{2} \)
Add 3 to both sides: \( y=\frac{3}{2}x-\frac{27}{2}+3=\frac{3}{2}x-\frac{27}{2}+\frac{6}{2}=\frac{3}{2}x-\frac{21}{2} \)

Step3: Recall slope of perpendicular lines

The slope of a line perpendicular to a line with slope \( m \) is \( m_{\text{perpendicular}}=-\frac{1}{m} \). For \( m = \frac{3}{2} \), \( m_{\text{perpendicular}}=-\frac{2}{3} \)

Step4: Use point - slope form for perpendicular line

Using point - slope form with \( (x_1,y_1)=(9,3) \) and \( m = -\frac{2}{3} \)
\( y - 3=-\frac{2}{3}(x - 9) \)
Expand: \( y - 3=-\frac{2}{3}x + 6 \)
Add 3 to both sides: \( y=-\frac{2}{3}x+6 + 3=-\frac{2}{3}x+9 \)

Answer:

Equation of parallel line: \( y=\frac{3}{2}x-\frac{21}{2} \)
Equation of perpendicular line: \( y = -\frac{2}{3}x+9 \)