Answer
To graph the equation \( x + 3y = 6 \), find the x-intercept at \( (6, 0) \) and the y-intercept at \( (0, 2) \). Plot these points and draw a straight line through them.
Solution
To graph the equation \( x + 3y = 6 \) using its intercepts, follow these steps:
### 1. Find the **x-intercept**
The **x-intercept** occurs where the graph crosses the x-axis. At this point, the value of \( y \) is **0**.
**Steps:**
- Set \( y = 0 \) in the equation:
\[
x + 3(0) = 6 \implies x = 6
\]
- **x-intercept:** \( (6, 0) \)
### 2. Find the **y-intercept**
The **y-intercept** occurs where the graph crosses the y-axis. At this point, the value of \( x \) is **0**.
**Steps:**
- Set \( x = 0 \) in the equation:
\[
0 + 3y = 6 \implies y = 2
\]
- **y-intercept:** \( (0, 2) \)
### 3. Plot the Intercepts
- Plot the **x-intercept** at \( (6, 0) \).
- Plot the **y-intercept** at \( (0, 2) \).
### 4. Draw the Graph
- Draw a straight line passing through the two plotted points \( (6, 0) \) and \( (0, 2) \).
- This line represents the equation \( x + 3y = 6 \).
### 5. Graphical Representation
Below is a simple graphical representation of the line:
```
y
|
2 | *
|
|
|
|
0 |------------------------ x
0 6
```
- The point \( (0, 2) \) is the **y-intercept** marked with an asterisk (*).
- The point \( (6, 0) \) is the **x-intercept**.
- The straight line connects these two intercepts.
### 6. Slope-Intercept Form (Optional)
For additional insight, you can rewrite the equation in **slope-intercept form** \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
**Steps:**
\[
x + 3y = 6 \\
3y = -x + 6 \\
y = -\frac{1}{3}x + 2
\]
- **Slope (\( m \))**: \( -\frac{1}{3} \)
- **y-intercept (\( b \))**: \( 2 \)
This form confirms that the line crosses the y-axis at \( (0, 2) \) and has a slope of \( -\frac{1}{3} \), meaning it decreases by 1 unit in \( y \) for every 3 units increase in \( x \).
### Summary
- **x-intercept:** \( (6, 0) \)
- **y-intercept:** \( (0, 2) \)
- **Slope-intercept form:** \( y = -\frac{1}{3}x + 2 \)
- **Graph:** A straight line passing through \( (6, 0) \) and \( (0, 2) \)
By following these steps, you can accurately graph the equation \( x + 3y = 6 \) using its intercepts.
Reviewed and approved by the UpStudy tutoring team
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