Let's solve this step-by-step:
1. Identify the slope of the given line:
The equation of 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. Determine the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope.
For the given line, the slope is [tex]\(-\frac{3}{4}\)[/tex].
The negative reciprocal of [tex]\(-\frac{3}{4}\)[/tex] is [tex]\(\frac{4}{3}\)[/tex].
Therefore, the slope of the perpendicular line is [tex]\(\frac{4}{3}\)[/tex].
3. Use the given x-intercept to find the equation:
We know that the x-intercept of the perpendicular line is 6. This means the line passes through the point [tex]\((6, 0)\)[/tex].
4. Form the equation using point-slope form:
The point-slope form of a line equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line.
Here, the slope [tex]\( m \)[/tex] is [tex]\(\frac{4}{3}\)[/tex] and the point [tex]\((x_1, y_1)\)[/tex] is [tex]\((6, 0)\)[/tex].
5. Substitute the values into the point-slope form:
[tex]\[
y - 0 = \frac{4}{3}(x - 6)
\][/tex]
6. Simplify the equation:
[tex]\[
y = \frac{4}{3} x - \frac{4}{3} \cdot 6
\][/tex]
[tex]\[
y = \frac{4}{3} x - 8
\][/tex]
So, the equation of the line that is perpendicular to the given line and has an [tex]\(x\)[/tex]-intercept of 6 is:
[tex]\[
y = \frac{4}{3} x - 8
\][/tex]
This corresponds to the third option:
[tex]\[
y = \frac{4}{3} x - 8
\][/tex]
