Question

consider the line $y = \frac{4}{5}x + 9$. find the equation of the line that is parallel to this line and passes through the point $(-4, 6)$. find the equation of the line that is perpendicular to this line and passes through the point $(-4, 6)$. note that the aleks graphing calculator may be helpful in checking your answer. equation of parallel line: $\square$ equation of perpendicular line: $\square$

Explanation:

Step1: Recall slope of parallel lines

Parallel lines have the same slope. The given line is \( y = \frac{4}{5}x + 9 \), so the slope (\( m \)) of the parallel line is also \( \frac{4}{5} \).

Step2: Use point - slope form for parallel line

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(-4,6) \) and \( m = \frac{4}{5} \).
Substitute the values: \( y - 6=\frac{4}{5}(x + 4) \)
Expand: \( y-6=\frac{4}{5}x+\frac{16}{5} \)
Add 6 to both sides: \( y=\frac{4}{5}x+\frac{16}{5}+6=\frac{4}{5}x+\frac{16 + 30}{5}=\frac{4}{5}x+\frac{46}{5} \)

Step3: Recall slope of perpendicular lines

The slope of a line perpendicular to a line with slope \( m \) is \( m_{\perp}=-\frac{1}{m} \). For \( m = \frac{4}{5} \), \( m_{\perp}=-\frac{5}{4} \)

Step4: Use point - slope form for perpendicular line

Using the point - slope form \( y - y_1 = m(x - x_1) \) with \( (x_1,y_1)=(-4,6) \) and \( m = -\frac{5}{4} \)
Substitute: \( y - 6=-\frac{5}{4}(x + 4) \)
Expand: \( y-6=-\frac{5}{4}x-5 \)
Add 6 to both sides: \( y=-\frac{5}{4}x-5 + 6=-\frac{5}{4}x+1 \)

Answer:

Equation of parallel line: \( y=\frac{4}{5}x+\frac{46}{5} \)
Equation of perpendicular line: \( y = -\frac{5}{4}x + 1 \)