Question

consider the line $y = -\frac{2}{3}x + 2,$
find the equation of the line that is parallel to this line and passes through the point $(-2, 5)$.
find the equation of the line that is perpendicular to this line and passes through the point $(-2, 5)$.
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 equal slopes. The given line is \( y = -\frac{2}{3}x + 2 \), so its slope \( m = -\frac{2}{3} \). The parallel line will also have \( m = -\frac{2}{3} \).

Step2: Use point - slope form for parallel line

Point - slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(-2,5) \) and \( m = -\frac{2}{3} \).
Substitute values: \( y - 5 = -\frac{2}{3}(x - (-2)) \)
Simplify: \( y - 5 = -\frac{2}{3}(x + 2) \)
Expand: \( y - 5 = -\frac{2}{3}x-\frac{4}{3} \)
Add 5 to both sides: \( y = -\frac{2}{3}x-\frac{4}{3}+\frac{15}{3} \)
So, \( y = -\frac{2}{3}x+\frac{11}{3} \)

Step3: Recall slope of perpendicular lines

Perpendicular lines have slopes that are negative reciprocals. If \( m_1 = -\frac{2}{3} \), then \( m_2=\frac{3}{2} \) (since \( m_1\times m_2=- 1\), \( -\frac{2}{3}\times\frac{3}{2}=-1 \))

Step4: Use point - slope form for perpendicular line

Using point - slope form with \( (x_1,y_1)=(-2,5) \) and \( m=\frac{3}{2} \)
\( y - 5=\frac{3}{2}(x - (-2)) \)
Simplify: \( y - 5=\frac{3}{2}(x + 2) \)
Expand: \( y - 5=\frac{3}{2}x + 3 \)
Add 5 to both sides: \( y=\frac{3}{2}x+3 + 5 \)
So, \( y=\frac{3}{2}x + 8 \)

Answer:

Equation of parallel line: \( y = -\frac{2}{3}x+\frac{11}{3} \)
Equation of perpendicular line: \( y=\frac{3}{2}x + 8 \)