Answer :
To find the equation of the line that is perpendicular to a given line and has a specific [tex]\( x \)[/tex]-intercept, we need to follow a few steps.
### Step 1: Understand the given equation
The given equations are:
1. [tex]\( y = -\frac{3}{4} x + 8 \)[/tex]
2. [tex]\( y = \frac{3}{4} x + 6 \)[/tex]
3. [tex]\( y = \frac{4}{3} x - 8 \)[/tex]
4. [tex]\( y = \frac{4}{3} x - 6 \)[/tex]
### Step 2: Determine the slope of the given line
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. For each equation:
- Equation 1: [tex]\( m = -\frac{3}{4} \)[/tex]
- Equation 2: [tex]\( m = \frac{3}{4} \)[/tex]
- Equation 3: [tex]\( m = \frac{4}{3} \)[/tex]
- Equation 4: [tex]\( m = \frac{4}{3} \)[/tex]
### Step 3: Find 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 [tex]\( m = -\frac{3}{4} \)[/tex], the perpendicular slope is [tex]\( \frac{4}{3} \)[/tex].
- For [tex]\( m = \frac{3}{4} \)[/tex], the perpendicular slope is [tex]\( -\frac{4}{3} \)[/tex].
- For [tex]\( m = \frac{4}{3} \)[/tex], the perpendicular slope is [tex]\( -\frac{3}{4} \)[/tex].
- For [tex]\( m = -\frac{4}{3} \)[/tex], the perpendicular slope is [tex]\( \frac{3}{4} \)[/tex].
### Step 4: Use the [tex]\( x \)[/tex]-intercept to find the y-intercept
Given an [tex]\( x \)[/tex]-intercept of 6, which means the line passes through the point [tex]\( (6, 0) \)[/tex].
Using the slope of [tex]\( \frac{4}{3} \)[/tex] (since the problem specifically states the perpendicular line has this slope as derived from the given line's perpendicular nature):
[tex]\[ y = mx + b \][/tex]
Substitute [tex]\( (6, 0) \)[/tex] into the equation:
[tex]\[ 0 = \frac{4}{3}(6) + b \][/tex]
[tex]\[ 0 = 8 + b \][/tex]
[tex]\[ b = -8 \][/tex]
So the y-intercept is [tex]\( -8 \)[/tex].
### Step 5: Write the equation in slope-intercept form
Now that we have both the slope and y-intercept:
[tex]\[ y = \frac{4}{3} x - 8 \][/tex]
Therefore, the correct equation is:
[tex]\[ y = \frac{4}{3} x - 8 \][/tex]
### Answer:
[tex]\[ \boxed{y = \frac{4}{3} x - 8} \][/tex]
### Step 1: Understand the given equation
The given equations are:
1. [tex]\( y = -\frac{3}{4} x + 8 \)[/tex]
2. [tex]\( y = \frac{3}{4} x + 6 \)[/tex]
3. [tex]\( y = \frac{4}{3} x - 8 \)[/tex]
4. [tex]\( y = \frac{4}{3} x - 6 \)[/tex]
### Step 2: Determine the slope of the given line
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. For each equation:
- Equation 1: [tex]\( m = -\frac{3}{4} \)[/tex]
- Equation 2: [tex]\( m = \frac{3}{4} \)[/tex]
- Equation 3: [tex]\( m = \frac{4}{3} \)[/tex]
- Equation 4: [tex]\( m = \frac{4}{3} \)[/tex]
### Step 3: Find 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 [tex]\( m = -\frac{3}{4} \)[/tex], the perpendicular slope is [tex]\( \frac{4}{3} \)[/tex].
- For [tex]\( m = \frac{3}{4} \)[/tex], the perpendicular slope is [tex]\( -\frac{4}{3} \)[/tex].
- For [tex]\( m = \frac{4}{3} \)[/tex], the perpendicular slope is [tex]\( -\frac{3}{4} \)[/tex].
- For [tex]\( m = -\frac{4}{3} \)[/tex], the perpendicular slope is [tex]\( \frac{3}{4} \)[/tex].
### Step 4: Use the [tex]\( x \)[/tex]-intercept to find the y-intercept
Given an [tex]\( x \)[/tex]-intercept of 6, which means the line passes through the point [tex]\( (6, 0) \)[/tex].
Using the slope of [tex]\( \frac{4}{3} \)[/tex] (since the problem specifically states the perpendicular line has this slope as derived from the given line's perpendicular nature):
[tex]\[ y = mx + b \][/tex]
Substitute [tex]\( (6, 0) \)[/tex] into the equation:
[tex]\[ 0 = \frac{4}{3}(6) + b \][/tex]
[tex]\[ 0 = 8 + b \][/tex]
[tex]\[ b = -8 \][/tex]
So the y-intercept is [tex]\( -8 \)[/tex].
### Step 5: Write the equation in slope-intercept form
Now that we have both the slope and y-intercept:
[tex]\[ y = \frac{4}{3} x - 8 \][/tex]
Therefore, the correct equation is:
[tex]\[ y = \frac{4}{3} x - 8 \][/tex]
### Answer:
[tex]\[ \boxed{y = \frac{4}{3} x - 8} \][/tex]
