To find the equation of the line that is perpendicular to a given line and has a specified [tex]\( x \)[/tex]-intercept, follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = -\frac{3}{4}x + 6 \)[/tex].
The slope 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.
Therefore, if the slope of the given line is [tex]\( -\frac{3}{4} \)[/tex], the slope [tex]\( m \)[/tex] of the perpendicular line will be:
[tex]\[
m = -\frac{1}{-\frac{3}{4}} = \frac{4}{3}
\][/tex]
3. Determine the equation of the perpendicular line:
We need a line with the slope [tex]\( \frac{4}{3} \)[/tex] and an [tex]\( x \)[/tex]-intercept of 6.
The general form of the equation of a line is:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
Since the [tex]\( x \)[/tex]-intercept is 6, at [tex]\( x = 6 \)[/tex], [tex]\( y = 0 \)[/tex]. Substituting these values into the equation:
[tex]\[
0 = \frac{4}{3} \cdot 6 + b
\][/tex]
4. Solve for the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]:
[tex]\[
0 = \frac{4}{3} \cdot 6 + b
\][/tex]
[tex]\[
0 = 8 + b
\][/tex]
[tex]\[
b = -8
\][/tex]
5. Write the final equation:
Substituting the slope [tex]\( \frac{4}{3} \)[/tex] and the [tex]\( y \)[/tex]-intercept [tex]\( -8 \)[/tex] back into the equation of the line, we get:
[tex]\[
y = \frac{4}{3}x - 8
\][/tex]
Therefore, 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]
The correct answer is:
[tex]\[ y = \frac{4}{3}x - 8 \][/tex]
