Question

consider the line y = -3x + 9. find the equation of the line that is perpendicular to this line and passes through the point (-8, 3). find the equation of the line that is parallel to this line and passes through the point (-8, 3). note that the aleks graphing calculator may be helpful in checking your answer. equation of perpendicular line: equation of parallel line:

Explanation:

Step1: Find slope of original line

The line \( y = -3x + 9 \) is in slope - intercept form \( y=mx + b \), where \( m \) is the slope. So the slope of the original line, \( m_{original}=-3 \).

Step2: Equation of perpendicular line

Step2.1: Find slope of perpendicular line

If two lines are perpendicular, the product of their slopes is \( - 1 \). Let the slope of the perpendicular line be \( m_{perp} \). Then \( m_{original}\times m_{perp}=-1 \). Substituting \( m_{original} = - 3 \), we get \( -3\times m_{perp}=-1 \), so \( m_{perp}=\frac{1}{3} \).

Step2.2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(-8,3) \) and \( m = \frac{1}{3} \).
Substituting the values, we have \( y - 3=\frac{1}{3}(x + 8) \).

Step2.3: Convert to slope - intercept form

\( y-3=\frac{1}{3}x+\frac{8}{3} \), then \( y=\frac{1}{3}x+\frac{8}{3}+3=\frac{1}{3}x+\frac{8 + 9}{3}=\frac{1}{3}x+\frac{17}{3} \).

Step3: Equation of parallel line

Step3.1: Find slope of parallel line

If two lines are parallel, their slopes are equal. So the slope of the parallel line, \( m_{para}=m_{original}=-3 \).

Step3.2: Use point - slope form

Using the point - slope form \( y - y_1=m(x - x_1) \) with \( (x_1,y_1)=(-8,3) \) and \( m=-3 \).
We get \( y - 3=-3(x + 8) \).

Step3.3: Convert to slope - intercept form

\( y-3=-3x-24 \), then \( y=-3x-24 + 3=-3x-21 \).

Answer:

Equation of perpendicular line: \( y=\frac{1}{3}x+\frac{17}{3} \)
Equation of parallel line: \( y=-3x - 21 \)