Question

find the slope of a line perpendicular to the line whose equation is 3x - 2y = 6. fully simplify your answer.

Explanation:

Step1: Convert to slope - intercept form

We start with the equation \(3x - 2y=6\). We want to solve for \(y\) to get it in the form \(y = mx + b\) (where \(m\) is the slope).
Subtract \(3x\) from both sides: \(- 2y=-3x + 6\)
Divide both sides by \(-2\): \(y=\frac{-3x + 6}{-2}=\frac{3}{2}x-3\)
So the slope of the given line \(m_1=\frac{3}{2}\)

Step2: Find the slope of the perpendicular line

If two lines are perpendicular, the product of their slopes \(m_1\times m_2=- 1\). Let \(m_2\) be the slope of the perpendicular line.
We know \(m_1 = \frac{3}{2}\), so \(\frac{3}{2}\times m_2=-1\)
Solve for \(m_2\): \(m_2=-1\div\frac{3}{2}=-1\times\frac{2}{3}=-\frac{2}{3}\)

Answer:

\(-\frac{2}{3}\)