To determine the equation of the line that is perpendicular to the given line and has an x-intercept of 6, follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = -\frac{3}{4}x + 8 \)[/tex]. The slope ([tex]\(m\)[/tex]) of this line is [tex]\(-\frac{3}{4}\)[/tex].
2. Find the slope of the perpendicular line:
Lines that are perpendicular to each other have slopes that are negative reciprocals. The negative reciprocal of [tex]\(-\frac{3}{4}\)[/tex] is [tex]\(\frac{4}{3}\)[/tex]. Thus, the slope of the perpendicular line is [tex]\(\frac{4}{3}\)[/tex].
3. Determine the x-intercept point:
The x-intercept of the new line is given as 6. This means the line passes through the point [tex]\((6, 0)\)[/tex].
4. Use the point-slope form to write the equation:
To find the equation of the line with a slope of [tex]\(\frac{4}{3}\)[/tex] passing through the point [tex]\((6, 0)\)[/tex], use the point-slope form [tex]\( y = mx + b \)[/tex].
- Substitute [tex]\(m = \frac{4}{3}\)[/tex], [tex]\(x = 6\)[/tex], and [tex]\(y = 0\)[/tex] into the equation:
[tex]\[
0 = \left(\frac{4}{3}\right) \cdot 6 + b
\][/tex]
5. Solve for the y-intercept ([tex]\(b\)[/tex]):
[tex]\[
0 = 8 + b
\][/tex]
[tex]\[
b = -8
\][/tex]
6. Write the equation of the line:
Now that we have the slope [tex]\(\frac{4}{3}\)[/tex] and the y-intercept [tex]\(-8\)[/tex], we can write the equation of the perpendicular line in slope-intercept form:
[tex]\[
y = \frac{4}{3}x - 8
\][/tex]
Therefore, the equation of the line that is perpendicular to the given line and has an x-intercept of 6 is:
[tex]\[
y = \frac{4}{3}x - 8
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{y = \frac{4}{3}x - 8}
\][/tex]
