Question

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

Explanation:

Step1: Recall slope of parallel lines

Parallel lines have equal slopes. The given line is \( y = \frac{2}{3}x + 2 \), so its slope \( m_{parallel}=\frac{2}{3} \).
Use point - slope form \( y - y_1 = m(x - x_1) \) with point \((-4,-2)\) and \( m=\frac{2}{3} \).
\( y - (-2)=\frac{2}{3}(x - (-4)) \)
\( y + 2=\frac{2}{3}(x + 4) \)

Step2: Simplify the parallel line equation

Expand the right - hand side: \( y+2=\frac{2}{3}x+\frac{8}{3} \)
Subtract 2 from both sides. Since \( 2=\frac{6}{3} \), we have \( y=\frac{2}{3}x+\frac{8}{3}-\frac{6}{3}=\frac{2}{3}x+\frac{2}{3} \)

Step3: Recall slope of perpendicular lines

The slope of a line perpendicular to a line with slope \( m \) is \( m_{perpendicular}=-\frac{1}{m} \). For \( m = \frac{2}{3} \), \( m_{perpendicular}=-\frac{3}{2} \)
Use point - slope form \( y - y_1=m(x - x_1) \) with point \((-4,-2)\) and \( m = -\frac{3}{2} \)
\( y-(-2)=-\frac{3}{2}(x - (-4)) \)
\( y + 2=-\frac{3}{2}(x + 4) \)

Step4: Simplify the perpendicular line equation

Expand the right - hand side: \( y + 2=-\frac{3}{2}x-6 \)
Subtract 2 from both sides: \( y=-\frac{3}{2}x-6 - 2=-\frac{3}{2}x-8 \)

Answer:

Equation of parallel line: \( y=\frac{2}{3}x+\frac{2}{3} \)
Equation of perpendicular line: \( y = -\frac{3}{2}x-8 \)