Question

consider the line $x - 3y = -8$. what is the slope of a line perpendicular to this line? what is the slope of a line parallel to this line? slope of a perpendicular line: $\square$ slope of a parallel line: $\square$

Explanation:

Step1: Rewrite the line in slope - intercept form

The equation of the line is \(x - 3y=-8\). We want to solve for \(y\) to get it in the form \(y = mx + b\), where \(m\) is the slope.
Subtract \(x\) from both sides: \(-3y=-x - 8\).
Divide each term by \(-3\): \(y=\frac{1}{3}x+\frac{8}{3}\). So the slope of the given line \(m=\frac{1}{3}\).

Step2: Find the slope of the parallel line

Parallel lines have the same slope. So the slope of a line parallel to the given line is also \(\frac{1}{3}\).

Step3: Find the slope of the 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 \(m\times m_{perp}=-1\).
We know \(m = \frac{1}{3}\), so \(\frac{1}{3}\times m_{perp}=-1\).
Solve for \(m_{perp}\) by multiplying both sides by \(3\): \(m_{perp}=-3\).

Answer:

Slope of a perpendicular line: \(-3\)
Slope of a parallel line: \(\frac{1}{3}\)