Question

consider the line -8x - 4y = 6. 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 given line

Rewrite \(-8x - 4y = 6\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
First, solve for \(y\):
\(-4y=8x + 6\)
Divide both sides by \(-4\): \(y=\frac{8x + 6}{-4}=-2x-\frac{3}{2}\)
So the slope of the given line \(m=-2\).

Step2: Equation of parallel line

Parallel lines have the same slope. So the slope of the parallel line \(m_{parallel}=-2\).
Use the point - slope form \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-4,-5)\) and \(m=-2\).
\(y-(-5)=-2(x - (-4))\)
\(y + 5=-2(x + 4)\)
\(y+5=-2x-8\)
\(y=-2x-13\) or \(2x + y=-13\)

Step3: Equation of perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is \(m_{perpendicular}=-\frac{1}{m}\).
Since \(m = - 2\), \(m_{perpendicular}=\frac{1}{2}\).
Use the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-4,-5)\) and \(m=\frac{1}{2}\).
\(y-(-5)=\frac{1}{2}(x - (-4))\)
\(y + 5=\frac{1}{2}(x + 4)\)
\(y+5=\frac{1}{2}x + 2\)
\(y=\frac{1}{2}x-3\) or \(x - 2y = 6\)

Answer:

Equation of parallel line: \(y=-2x - 13\) (or \(2x+y=-13\))
Equation of perpendicular line: \(y=\frac{1}{2}x-3\) (or \(x - 2y = 6\))