Question

consider the line -3x + 2y = 8.
find the equation of the line that is parallel to this line and passes through the point (4, -5).
find the equation of the line that is perpendicular to this line and passes through the point (4, -5).
note that the aleks graphing calculator may be helpful in checking your answer.
equation of parallel line:
equation of perpendicular line:

Explanation:

Step1: Find slope of original line

Rewrite \(-3x + 2y = 8\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
Add \(3x\) to both sides: \(2y=3x + 8\).
Divide by 2: \(y=\frac{3}{2}x + 4\). So the slope \(m\) of the original line is \(\frac{3}{2}\).

Step2: Equation of parallel line

Parallel lines have the same slope. So the slope of the parallel line \(m_{parallel}=\frac{3}{2}\).
Use the point - slope form \(y - y_1=m(x - x_1)\) where \((x_1,y_1)=(4,-5)\) and \(m = \frac{3}{2}\).
\(y-(-5)=\frac{3}{2}(x - 4)\)
Simplify: \(y + 5=\frac{3}{2}x-6\)
Subtract 5: \(y=\frac{3}{2}x-11\). Multiply through by 2 to get standard form: \(2y = 3x-22\) or \(3x-2y = 22\).

Step3: Equation of perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is \(m_{perpendicular}=-\frac{1}{m}\). So \(m_{perpendicular}=-\frac{2}{3}\).
Use point - slope form with \((x_1,y_1)=(4,-5)\) and \(m = -\frac{2}{3}\).
\(y-(-5)=-\frac{2}{3}(x - 4)\)
Simplify: \(y + 5=-\frac{2}{3}x+\frac{8}{3}\)
Subtract 5 (which is \(\frac{15}{3}\)): \(y=-\frac{2}{3}x+\frac{8}{3}-\frac{15}{3}=-\frac{2}{3}x-\frac{7}{3}\). Multiply through by 3: \(3y=-2x - 7\) or \(2x+3y=-7\).

Answer:

Equation of parallel line: \(y=\frac{3}{2}x - 11\) (or \(3x - 2y=22\))
Equation of perpendicular line: \(y = -\frac{2}{3}x-\frac{7}{3}\) (or \(2x + 3y=-7\))