Answer
Simplify the equation \( 3y - 2x = 6 \) to \( y = \frac{2}{3}x + 2 \). Find the \( y \)-intercept at \( (0, 2) \) and the \( x \)-intercept at \( (-3, 0) \). Graph the line by plotting these intercepts and drawing a straight line through them.
Solution
Sure! Let's simplify the given equation, find the intercepts, and then outline how to graph it.
### Given Equation:
\[ 3y - 2x = 6 \]
### Step 1: Simplify the Equation
We'll solve for \( y \) to get the equation into slope-intercept form (\( y = mx + b \)).
\[
\begin{align*}
3y - 2x &= 6 \\
3y &= 2x + 6 \quad &\text{(Add \( 2x \) to both sides)} \\
y &= \frac{2}{3}x + 2 \quad &\text{(Divide both sides by 3)}
\end{align*}
\]
**Simplified Equation:**
\[ y = \frac{2}{3}x + 2 \]
### Step 2: Find the Intercepts
**1. \( y \)-Intercept:**
The \( y \)-intercept occurs where \( x = 0 \).
\[
y = \frac{2}{3}(0) + 2 = 0 + 2 = 2
\]
\[
\text{\( y \)-intercept: } (0, 2)
\]
**2. \( x \)-Intercept:**
The \( x \)-intercept occurs where \( y = 0 \).
\[
0 = \frac{2}{3}x + 2 \\
\frac{2}{3}x = -2 \\
x = -2 \times \frac{3}{2} = -3
\]
\[
\text{\( x \)-intercept: } (-3, 0)
\]
### Step 3: Graph the Equation
To graph the equation \( y = \frac{2}{3}x + 2 \), follow these steps:
1. **Plot the Intercepts:**
- **\( y \)-intercept:** Plot the point \( (0, 2) \).
- **\( x \)-intercept:** Plot the point \( (-3, 0) \).
2. **Draw the Line:**
- Use a ruler to draw a straight line passing through both intercepts.
- Extend the line across the graph and add arrows on both ends to indicate that it continues indefinitely.
3. **Label the Line:**
- Optionally, label the line with its equation \( y = \frac{2}{3}x + 2 \).
### Graph Visualization
Since I can't produce visual images directly, here's a simple way to visualize the graph:
```
Y
|
| *
| (0,2)
|
|
|
|
|-------------------------- X
|
|
|
|
| *
| (-3,0)
```
- The point \( (0, 2) \) is the \( y \)-intercept.
- The point \( (-3, 0) \) is the \( x \)-intercept.
- The straight line connects these two points, representing all solutions to the equation \( 3y - 2x = 6 \).
### Using a Graphing Tool
You can use various online graphing tools like [Desmos](https://www.desmos.com/calculator) or [GeoGebra](https://www.geogebra.org/graphing) to plot the equation:
1. **Open the Graphing Tool:** Navigate to your preferred graphing website.
2. **Enter the Equation:** Type in \( 3y - 2x = 6 \) or the simplified form \( y = \frac{2}{3}x + 2 \).
3. **View the Graph:** The tool will display the line with the intercepts marked.
4. **Verify the Intercepts:** Ensure that the graph passes through \( (-3, 0) \) and \( (0, 2) \).
### Summary
- **Simplified Equation:** \( y = \frac{2}{3}x + 2 \)
- **\( y \)-Intercept:** \( (0, 2) \)
- **\( x \)-Intercept:** \( (-3, 0) \)
- **Graph:** A straight line passing through both intercepts.
Feel free to use graphing paper or an online tool to visualize the equation accurately!
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
