Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Ask your questions and receive precise answers from experienced professionals across different disciplines. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
