To find the slope of a line perpendicular to the line given by the equation [tex]\(6x - 9y = 216\)[/tex], follow these steps:
1. Convert the equation to slope-intercept form ( [tex]\( y = mx + b \)[/tex] ):
- Start with the original equation: [tex]\(6x - 9y = 216\)[/tex].
- Solve for [tex]\(y\)[/tex] by isolating it on one side of the equation.
[tex]\[
6x - 9y = 216
\][/tex]
Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
-9y = -6x + 216
\][/tex]
Divide every term by [tex]\(-9\)[/tex]:
[tex]\[
y = \left(\frac{-6}{-9}\right)x + \left(\frac{216}{-9}\right)
\][/tex]
Simplify the fractions:
[tex]\[
y = \left(\frac{2}{3}\right)x - 24
\][/tex]
So, the equation in slope-intercept form is [tex]\(y = \frac{2}{3}x - 24\)[/tex].
2. Identify the slope of the given line:
- From the equation [tex]\(y = \frac{2}{3}x - 24\)[/tex], the slope [tex]\(m\)[/tex] of the line is [tex]\(\frac{2}{3}\)[/tex].
3. Find the slope of the perpendicular line:
- The slope of a line perpendicular to another line is the negative reciprocal of the original slope.
- The original slope is [tex]\(\frac{2}{3}\)[/tex].
- The negative reciprocal of [tex]\(\frac{2}{3}\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex].
Therefore, the slope of the line perpendicular to the line [tex]\(6x - 9y = 216\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex].
