What is the equation of the line that is perpendicular to the given line and has an [tex]$x$[/tex]-intercept of 6?

A. [tex]\( y = -\frac{3}{4} x + 8 \)[/tex]

B. [tex]\( y = \frac{3}{4} x + 6 \)[/tex]

C. [tex]\( y = \frac{4}{3} x - 8 \)[/tex]

D. [tex]\( y = \frac{4}{3} x - 6 \)[/tex]



Answer :

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]