To determine the equation of the line that is perpendicular to the given line \( y = -\frac{3}{4} x + b \) and has an \( x \)-intercept of 6, follow these detailed steps:
Step 1: Identify the slope of the given line.
The given line is in the form \( y = mx + b \), where \( m \) is the slope. For the given line \( y = -\frac{3}{4} x + b \), the slope \( m \) is \( -\frac{3}{4} \).
Step 2: Determine the slope of the perpendicular line.
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \( -\frac{3}{4} \) is \( \frac{4}{3} \).
Step 3: Use the \( x \)-intercept to find the y-intercept.
The \( x \)-intercept of a line is the point where the line crosses the \( x \)-axis. At this point, \( y = 0 \). Given the \( x \)-intercept is 6, the line passes through the point \( (6, 0) \).
Step 4: Substitute the values into the line equation to solve for \( b \).
The general form of the equation of the line is \( y = mx + b \). Using the slope \( m = \frac{4}{3} \) and the point \( (6, 0) \), substitute these values into the equation to solve for \( b \):
[tex]\[
0 = \left( \frac{4}{3} \right) \cdot 6 + b
\][/tex]
[tex]\[
0 = 8 + b
\][/tex]
[tex]\[
b = -8
\][/tex]
Step 5: Write the equation of the line.
Now that we have the slope \( \frac{4}{3} \) and the y-intercept \( b = -8 \), the equation of the line is:
[tex]\[
y = \frac{4}{3} x - 8
\][/tex]
Identify the correct option:
From the given options, the correct equation that matches our result is:
[tex]\[
y = \frac{4}{3} x - 8
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
