Question

Write y - 2 = frac{3}{2}(x + 2) in slope-intercept form..
Answer
Graph the equation 9 - x = frac{3}{2}y using slope-intercept form.
Answer
Graph the equation 3x - y = 10 using slope-intercept form.
Answer
Identify the slope and the y-intercept of the equation 4x - 3y = 12
Answered by GPT-4o mini
To convert the equation \( y - 2 = \frac{3}{2}(x + 2) \) into slope-intercept form (which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept), we will follow these steps:

1. Distribute \(\frac{3}{2}\) on the right side:
\[
y - 2 = \frac{3}{2}x + \frac{3}{2} \cdot 2
\]
\[
y - 2 = \frac{3}{2}x + 3
\]

2. Now, add \(2\) to both sides to isolate \(y\):
\[
y = \frac{3}{2}x + 3 + 2
\]
\[
y = \frac{3}{2}x + 5
\]

Now the equation is in slope-intercept form:
\[
y = \frac{3}{2}x + 5
\]

In this form, the slope \(m\) is \(\frac{3}{2}\) and the y-intercept \(b\) is \(5\).
Answered by GPT-4o mini
To graph the equation \(9 - x = \frac{3}{2}y\), we first need to convert it into slope-intercept form (\(y = mx + b\)).

### Step 1: Rearrange the equation

Starting with the original equation:
\[
9 - x = \frac{3}{2}y
\]

We can rearrange it to isolate \(y\):

1. Move \(x\) to the right side:
\[
\frac{3}{2}y = 9 - x
\]

2. Now, solve for \(y\) by multiplying both sides by \(\frac{2}{3}\) in order to isolate \(y\):
\[
y = \frac{2}{3}(9 - x)
\]

3. Distributing \(\frac{2}{3}\):
\[
y = \frac{2}{3} \cdot 9 - \frac{2}{3}x
\]
\[
y = 6 - \frac{2}{3}x
\]

Now we have the equation in slope-intercept form:
\[
y = -\frac{2}{3}x + 6
\]

### Step 2: Identify the slope and intercept

- **Slope (m)**: \(-\frac{2}{3}\)
- **Y-intercept (b)**: \(6\) (this means the line crosses the y-axis at \( (0, 6) \))

### Step 3: Plot the y-intercept

Start by plotting the y-intercept on the graph:
- Point: \( (0, 6) \)

### Step 4: Use the slope to find another point

From the y-intercept \( (0, 6) \), use the slope \(-\frac{2}{3}\) to find another point:
- This means for every \(3\) units you move to the right (positive direction along the x-axis), you move \(2\) units down (negative direction along the y-axis).

Starting from \( (0, 6) \):
1. Move \(3\) units to the right: \(x = 0 + 3 = 3\)
2. Move \(2\) units down: \(y = 6 - 2 = 4\)

So, another point on the line is \( (3, 4) \).

### Step 5: Plot the second point

Now you can plot the point \( (3, 4) \) on the graph.

### Step 6: Draw the line

Draw a straight line through the points \( (0, 6) \) and \( (3, 4) \). Extend the line in both directions, and make sure to add arrows on both ends to indicate that the line continues indefinitely.

### Summary of the points:

- **Y-intercept**: \( (0, 6) \)
- **Another point**: \( (3, 4) \)

These will create a line with a negative slope, indicating that as \(x\) increases, \(y\) decreases.

If you were to graph this on an actual coordinate plane, you would see the line descending from left to right, consistent with the negative slope.
Answered by GPT-4o mini
To graph the equation \(3x - y = 10\) using slope-intercept form, we will first convert it into the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

### Step 1: Rearrange the equation

Starting with the original equation:
\[
3x - y = 10
\]
We can solve for \(y\):

1. Subtract \(3x\) from both sides:
\[
-y = -3x + 10
\]

2. Multiply both sides by \(-1\) to isolate \(y\):
\[
y = 3x - 10
\]

Now we have the equation in slope-intercept form:
\[
y = 3x - 10
\]

### Step 2: Identify the slope and intercept

- **Slope (m)**: \(3\)
- **Y-intercept (b)**: \(-10\)

### Step 3: Plot the y-intercept

Start by plotting the y-intercept on the graph:
- The y-intercept \((0, -10)\) means that when \(x = 0\), \(y = -10\).

### Step 4: Use the slope to find another point

From the y-intercept \((0, -10)\), use the slope \(3\):
- The slope is \(3\), which can be expressed as \(\frac{3}{1}\). This means that for every \(1\) unit you move to the right (in the positive direction), you move \(3\) units up (in the positive direction).

Starting from \((0, -10)\):
1. Move \(1\) unit to the right: \(x = 0 + 1 = 1\)
2. Move \(3\) units up: \(y = -10 + 3 = -7\)

So, another point on the line is \((1, -7)\).

### Step 5: Plot the second point

Now you can plot the point \((1, -7)\) on the graph.

### Step 6: Draw the line

Draw a straight line through the points \((0, -10)\) and \((1, -7)\). Extend the line in both directions, and add arrows on both ends to indicate that the line continues indefinitely.

### Summary of the points:

- **Y-intercept**: \((0, -10)\)
- **Another point**: \((1, -7)\)

This line will have a positive slope, indicating that as \(x\) increases, \(y\) also increases.

### Graph

If you were to graph this on coordinate paper or a tool, you'd see that the line rises steeply from left to right due to the slope of \(3\).
Answered by GPT-4o mini
To identify the slope and the y-intercept of the equation \(4x - 3y = 12\), we need to rewrite the equation in slope-intercept form, which is given by \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.

### Step 1: Rearrange the equation

Starting with the original equation:
\[
4x - 3y = 12
\]
We can isolate \(y\) by following these steps:

1. Subtract \(4x\) from both sides:
\[
-3y = -4x + 12
\]

2. Divide every term by \(-3\) to solve for \(y\):
\[
y = \frac{4}{3}x - 4
\]

### Step 2: Identify the slope and the y-intercept

Now that we have the equation in slope-intercept form \(y = \frac{4}{3}x - 4\):

- **Slope (m)**: \(\frac{4}{3}\)
- **Y-intercept (b)**: \(-4\)

### Summary

- The slope of the equation \(4x - 3y = 12\) is \(\frac{4}{3}\).
- The y-intercept is \(-4\) (which corresponds to the point \((0, -4)\) on the graph).