To find the slope of a line perpendicular to the line given by the equation [tex]\(6x + 8y = -128\)[/tex], we need to follow these steps:
1. Rewrite the equation in slope-intercept form ([tex]\(y = mx + b\)[/tex]).
Start with the given equation:
[tex]\[
6x + 8y = -128
\][/tex]
Isolate [tex]\(y\)[/tex] by moving [tex]\(6x\)[/tex] to the other side:
[tex]\[
8y = -6x - 128
\][/tex]
Divide every term by 8 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-6}{8}x - \frac{128}{8}
\][/tex]
Simplify the fractions:
[tex]\[
y = -\frac{3}{4}x - 16
\][/tex]
2. Identify the slope ([tex]\(m\)[/tex]) of the given line.
From the equation [tex]\(y = -\frac{3}{4}x - 16\)[/tex], we can see that the slope [tex]\(m\)[/tex] of the given line is:
[tex]\[
m = -\frac{3}{4}
\][/tex]
3. Find the slope of the line perpendicular to the given line.
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. To find the negative reciprocal, we take the reciprocal of [tex]\(-\frac{3}{4}\)[/tex] and change its sign.
The reciprocal of [tex]\(-\frac{3}{4}\)[/tex] is [tex]\(-\frac{4}{3}\)[/tex]. Changing its sign gives us:
[tex]\[
\text{slope of the perpendicular line} = \frac{4}{3}
\][/tex]
Therefore, the slope of a line perpendicular to the line given by [tex]\(6x + 8y = -128\)[/tex] is:
[tex]\[
\frac{4}{3}
\][/tex]
