Question
Write an equation of the line satisfying the given conditions. (Hint: Recall the relationships among slopes of perpendicular lines.)
Perpendicular to xnegative 2yequals9; y-intercept (0,negative 3)
Perpendicular to xnegative 2yequals9; y-intercept (0,negative 3)
Answers
GPT-4o mini
To find the equation of the line that is perpendicular to the line given by \( x - 2y = 9 \) and has a y-intercept of \( (0, -3) \), we will follow these steps:
1. **Convert the equation to slope-intercept form (y = mx + b)** to find the slope of the given line.
2. **Determine the slope of the perpendicular line.**
3. **Use the slope and the y-intercept point to write the equation of the new line.**
### Step 1: Convert \( x - 2y = 9 \) to slope-intercept form
Start with the original equation:
\[
x - 2y = 9
\]
Rearranging it to isolate \( y \):
\[
-2y = -x + 9
\]
Now, divide by -2:
\[
y = \frac{1}{2}x - \frac{9}{2}
\]
From this, we see that the slope \( m \) of the given line is \( \frac{1}{2} \).
### Step 2: Determine the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Thus, for the slope \( \frac{1}{2} \):
\[
m_{\text{perpendicular}} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\]
### Step 3: Write the equation of the new line
Now, we know the y-intercept is \( -3 \) (which means the point is \( (0, -3) \)), and the slope of the line we want to write is \( -2 \).
Using the slope-intercept form \( y = mx + b \):
\[
y = -2x - 3
\]
This is the equation of the line that is perpendicular to \( x - 2y = 9 \) and has a y-intercept of \( (0, -3) \).
1. **Convert the equation to slope-intercept form (y = mx + b)** to find the slope of the given line.
2. **Determine the slope of the perpendicular line.**
3. **Use the slope and the y-intercept point to write the equation of the new line.**
### Step 1: Convert \( x - 2y = 9 \) to slope-intercept form
Start with the original equation:
\[
x - 2y = 9
\]
Rearranging it to isolate \( y \):
\[
-2y = -x + 9
\]
Now, divide by -2:
\[
y = \frac{1}{2}x - \frac{9}{2}
\]
From this, we see that the slope \( m \) of the given line is \( \frac{1}{2} \).
### Step 2: Determine the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Thus, for the slope \( \frac{1}{2} \):
\[
m_{\text{perpendicular}} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\]
### Step 3: Write the equation of the new line
Now, we know the y-intercept is \( -3 \) (which means the point is \( (0, -3) \)), and the slope of the line we want to write is \( -2 \).
Using the slope-intercept form \( y = mx + b \):
\[
y = -2x - 3
\]
This is the equation of the line that is perpendicular to \( x - 2y = 9 \) and has a y-intercept of \( (0, -3) \).
Are you sure?
GPT-4o mini
Let me double-check the calculations and steps to ensure everything is correct.
### 1. Given Line: \( x - 2y = 9 \)
**Convert to Slope-Intercept Form:**
Start with the equation:
\[
x - 2y = 9
\]
Rearranging it for \( y \):
\[
-2y = -x + 9
\]
Now divide by -2:
\[
y = \frac{1}{2}x - \frac{9}{2}
\]
The slope of this line is \( m = \frac{1}{2} \).
### 2. Find the Perpendicular Slope
To find the slope of the line that is perpendicular to this line, take the negative reciprocal of \( \frac{1}{2} \):
\[
m_{\text{perpendicular}} = -\frac{1}{\frac{1}{2}} = -2
\]
### 3. Use the Y-Intercept to Write the Equation
Given that the line has a y-intercept of \( (0, -3) \), we can use the slope-intercept form \( y = mx + b \). Here, \( m = -2 \) and \( b = -3 \):
\[
y = -2x - 3
\]
### Conclusion
So, the final equation, \( y = -2x - 3 \), is indeed correct. It represents a line that is perpendicular to the original line \( x - 2y = 9 \) and passes through the point \( (0, -3) \).
If you have any specific concerns or additional details, please let me know!
### 1. Given Line: \( x - 2y = 9 \)
**Convert to Slope-Intercept Form:**
Start with the equation:
\[
x - 2y = 9
\]
Rearranging it for \( y \):
\[
-2y = -x + 9
\]
Now divide by -2:
\[
y = \frac{1}{2}x - \frac{9}{2}
\]
The slope of this line is \( m = \frac{1}{2} \).
### 2. Find the Perpendicular Slope
To find the slope of the line that is perpendicular to this line, take the negative reciprocal of \( \frac{1}{2} \):
\[
m_{\text{perpendicular}} = -\frac{1}{\frac{1}{2}} = -2
\]
### 3. Use the Y-Intercept to Write the Equation
Given that the line has a y-intercept of \( (0, -3) \), we can use the slope-intercept form \( y = mx + b \). Here, \( m = -2 \) and \( b = -3 \):
\[
y = -2x - 3
\]
### Conclusion
So, the final equation, \( y = -2x - 3 \), is indeed correct. It represents a line that is perpendicular to the original line \( x - 2y = 9 \) and passes through the point \( (0, -3) \).
If you have any specific concerns or additional details, please let me know!