Question
Transform the linear equation 4x−8y=24 to slope-intercept form.(1 point)
Write y−2=3/2(x+2) in slope-intercept form.(1 point)
Graph the equation 9−x=3/2y using slope-intercept form.(1 point)
Write y−2=3/2(x+2) in slope-intercept form.(1 point)
Graph the equation 9−x=3/2y using slope-intercept form.(1 point)
Answers
GPT-4o mini
Let's address each part of your request step-by-step.
### 1. Transform the linear equation \(4x - 8y = 24\) to slope-intercept form.
The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Starting with the equation:
\[
4x - 8y = 24
\]
First, isolate \(y\):
\[
-8y = -4x + 24
\]
Now, divide every term by \(-8\):
\[
y = \frac{1}{2}x - 3
\]
Thus, the slope-intercept form is:
\[
y = \frac{1}{2}x - 3
\]
### 2. Write \(y - 2 = \frac{3}{2}(x + 2)\) in slope-intercept form.
Start with the equation:
\[
y - 2 = \frac{3}{2}(x + 2)
\]
Distribute \(\frac{3}{2}\) on the right side:
\[
y - 2 = \frac{3}{2}x + 3
\]
Now, add \(2\) to both sides to solve for \(y\):
\[
y = \frac{3}{2}x + 3 + 2
\]
\[
y = \frac{3}{2}x + 5
\]
Thus, the slope-intercept form is:
\[
y = \frac{3}{2}x + 5
\]
### 3. Graph the equation \(9 - x = \frac{3}{2}y\) using slope-intercept form.
Start with the equation:
\[
9 - x = \frac{3}{2}y
\]
Rearranging to isolate \(y\):
\[
\frac{3}{2}y = 9 - x
\]
Multiply through by \(\frac{2}{3}\):
\[
y = \frac{2}{3}(9 - x)
\]
\[
y = 6 - \frac{2}{3}x
\]
Now, the slope-intercept form is:
\[
y = -\frac{2}{3}x + 6
\]
### Graphing the Equation
To graph \(y = -\frac{2}{3}x + 6\):
1. **Identify the y-intercept**: \(b = 6\). This is the point \((0, 6)\) on the y-axis.
2. **Identify the slope**: \(m = -\frac{2}{3}\). This means for every 3 units you move to the right (positive x-direction), you move down 2 units (negative y-direction).
**Plotting the points**:
- Start at the y-intercept \((0, 6)\).
- From \((0, 6)\), move 3 units to the right to \((3, 6)\) and then down 2 units to the point \((3, 4)\).
- Draw a straight line through these points, extending it in both directions.
Now you've transformed the equations and know how to graph the last one!
### 1. Transform the linear equation \(4x - 8y = 24\) to slope-intercept form.
The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Starting with the equation:
\[
4x - 8y = 24
\]
First, isolate \(y\):
\[
-8y = -4x + 24
\]
Now, divide every term by \(-8\):
\[
y = \frac{1}{2}x - 3
\]
Thus, the slope-intercept form is:
\[
y = \frac{1}{2}x - 3
\]
### 2. Write \(y - 2 = \frac{3}{2}(x + 2)\) in slope-intercept form.
Start with the equation:
\[
y - 2 = \frac{3}{2}(x + 2)
\]
Distribute \(\frac{3}{2}\) on the right side:
\[
y - 2 = \frac{3}{2}x + 3
\]
Now, add \(2\) to both sides to solve for \(y\):
\[
y = \frac{3}{2}x + 3 + 2
\]
\[
y = \frac{3}{2}x + 5
\]
Thus, the slope-intercept form is:
\[
y = \frac{3}{2}x + 5
\]
### 3. Graph the equation \(9 - x = \frac{3}{2}y\) using slope-intercept form.
Start with the equation:
\[
9 - x = \frac{3}{2}y
\]
Rearranging to isolate \(y\):
\[
\frac{3}{2}y = 9 - x
\]
Multiply through by \(\frac{2}{3}\):
\[
y = \frac{2}{3}(9 - x)
\]
\[
y = 6 - \frac{2}{3}x
\]
Now, the slope-intercept form is:
\[
y = -\frac{2}{3}x + 6
\]
### Graphing the Equation
To graph \(y = -\frac{2}{3}x + 6\):
1. **Identify the y-intercept**: \(b = 6\). This is the point \((0, 6)\) on the y-axis.
2. **Identify the slope**: \(m = -\frac{2}{3}\). This means for every 3 units you move to the right (positive x-direction), you move down 2 units (negative y-direction).
**Plotting the points**:
- Start at the y-intercept \((0, 6)\).
- From \((0, 6)\), move 3 units to the right to \((3, 6)\) and then down 2 units to the point \((3, 4)\).
- Draw a straight line through these points, extending it in both directions.
Now you've transformed the equations and know how to graph the last one!
bigepic
how did u get three when u distibuted the 2nd one
GPT-4o mini
Let’s carefully go through the distribution step in the equation \(y - 2 = \frac{3}{2}(x + 2)\).
### Starting Equation
\[
y - 2 = \frac{3}{2}(x + 2)
\]
### Step 1: Distributing \(\frac{3}{2}\)
When you distribute \(\frac{3}{2}\) across the quantity \((x + 2)\), you need to apply the multiplication separately to both \(x\) and \(2\):
\[
y - 2 = \frac{3}{2} \cdot x + \frac{3}{2} \cdot 2
\]
Calculating each part:
1. \(\frac{3}{2} \cdot x = \frac{3}{2}x\)
2. \(\frac{3}{2} \cdot 2 = 3\) (since \(2 = \frac{2}{1}\) and \(\frac{3 \cdot 2}{2 \cdot 1} = \frac{6}{2} = 3\))
So the distribution outputs:
\[
y - 2 = \frac{3}{2}x + 3
\]
### Step 2: Solving for \(y\)
Now you add \(2\) to both sides to isolate \(y\):
\[
y = \frac{3}{2}x + 3 + 2
\]
Combine the constants:
\[
y = \frac{3}{2}x + 5
\]
### Final Result
The slope-intercept form is:
\[
y = \frac{3}{2}x + 5
\]
Thank you for pointing out the need for clarity; I hope this explanation clarifies the distribution step!
### Starting Equation
\[
y - 2 = \frac{3}{2}(x + 2)
\]
### Step 1: Distributing \(\frac{3}{2}\)
When you distribute \(\frac{3}{2}\) across the quantity \((x + 2)\), you need to apply the multiplication separately to both \(x\) and \(2\):
\[
y - 2 = \frac{3}{2} \cdot x + \frac{3}{2} \cdot 2
\]
Calculating each part:
1. \(\frac{3}{2} \cdot x = \frac{3}{2}x\)
2. \(\frac{3}{2} \cdot 2 = 3\) (since \(2 = \frac{2}{1}\) and \(\frac{3 \cdot 2}{2 \cdot 1} = \frac{6}{2} = 3\))
So the distribution outputs:
\[
y - 2 = \frac{3}{2}x + 3
\]
### Step 2: Solving for \(y\)
Now you add \(2\) to both sides to isolate \(y\):
\[
y = \frac{3}{2}x + 3 + 2
\]
Combine the constants:
\[
y = \frac{3}{2}x + 5
\]
### Final Result
The slope-intercept form is:
\[
y = \frac{3}{2}x + 5
\]
Thank you for pointing out the need for clarity; I hope this explanation clarifies the distribution step!
bigepic
graph 3x−y=10
GPT-4o mini
To graph the equation \(3x - y = 10\), we'll first rewrite it in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
### Step 1: Rewrite the equation
Starting with the original equation:
\[
3x - y = 10
\]
Isolate \(y\):
\[
-y = -3x + 10
\]
\[
y = 3x - 10
\]
Now we have it in slope-intercept form:
\[
y = 3x - 10
\]
### Step 2: Identify the slope and y-intercept
From the equation \(y = 3x - 10\):
- The slope \(m = 3\)
- The y-intercept \(b = -10\)
This means that the line crosses the y-axis at the point \((0, -10)\).
### Step 3: Plot the y-intercept
1. Plot the point \((0, -10)\) on the Cartesian plane.
### Step 4: Use the slope to find another point
Since the slope is \(3\) (or \(\frac{3}{1}\)), you can go up \(3\) units and then over \(1\) unit to the right:
From the point \((0, -10)\):
- Move up \(3\) to \(y = -7\), and move \(1\) unit right to \(x = 1\), giving you the point \((1, -7)\).
### Step 5: Plot the second point
Plot the point \((1, -7)\).
### Step 6: Draw the line
Now, draw a straight line through the points \((0, -10)\) and \((1, -7)\), extending the line in both directions. You can also find additional points if desired by using the slope again or using other values for \(x\).
### Additional Points (Optional)
To find a few more points:
- For \(x = 2\):
\[
y = 3(2) - 10 = 6 - 10 = -4 \quad \Rightarrow \quad (2, -4)
\]
- For \(x = -2\):
\[
y = 3(-2) - 10 = -6 - 10 = -16 \quad \Rightarrow \quad (-2, -16)
\]
### Final Result
The graph of the equation \(3x - y = 10\) is a straight line that crosses the y-axis at \((0, -10)\) and has a slope of \(3\), meaning it rises \(3\) units for every \(1\) unit it moves to the right.
If you were to visualize this on graph paper or using a graphing tool, you would see a line passing through those points.
### Step 1: Rewrite the equation
Starting with the original equation:
\[
3x - y = 10
\]
Isolate \(y\):
\[
-y = -3x + 10
\]
\[
y = 3x - 10
\]
Now we have it in slope-intercept form:
\[
y = 3x - 10
\]
### Step 2: Identify the slope and y-intercept
From the equation \(y = 3x - 10\):
- The slope \(m = 3\)
- The y-intercept \(b = -10\)
This means that the line crosses the y-axis at the point \((0, -10)\).
### Step 3: Plot the y-intercept
1. Plot the point \((0, -10)\) on the Cartesian plane.
### Step 4: Use the slope to find another point
Since the slope is \(3\) (or \(\frac{3}{1}\)), you can go up \(3\) units and then over \(1\) unit to the right:
From the point \((0, -10)\):
- Move up \(3\) to \(y = -7\), and move \(1\) unit right to \(x = 1\), giving you the point \((1, -7)\).
### Step 5: Plot the second point
Plot the point \((1, -7)\).
### Step 6: Draw the line
Now, draw a straight line through the points \((0, -10)\) and \((1, -7)\), extending the line in both directions. You can also find additional points if desired by using the slope again or using other values for \(x\).
### Additional Points (Optional)
To find a few more points:
- For \(x = 2\):
\[
y = 3(2) - 10 = 6 - 10 = -4 \quad \Rightarrow \quad (2, -4)
\]
- For \(x = -2\):
\[
y = 3(-2) - 10 = -6 - 10 = -16 \quad \Rightarrow \quad (-2, -16)
\]
### Final Result
The graph of the equation \(3x - y = 10\) is a straight line that crosses the y-axis at \((0, -10)\) and has a slope of \(3\), meaning it rises \(3\) units for every \(1\) unit it moves to the right.
If you were to visualize this on graph paper or using a graphing tool, you would see a line passing through those points.