Question
Graph the equation using the slope and the \( y \)-intercept. \( x+2 y=6 \)
Ask by Ramirez Hall. in the United States
Feb 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To graph the equation \( x + 2y = 6 \), follow these steps:
1. **Convert to Slope-Intercept Form**:
- Start with \( x + 2y = 6 \).
- Subtract \( x \) from both sides: \( 2y = 6 - x \).
- Divide by 2: \( y = 3 - \frac{1}{2}x \).
2. **Identify Slope and Y-Intercept**:
- Slope (\( m \)) is \( -\frac{1}{2} \).
- Y-intercept (\( b \)) is 3, so the point is (0, 3).
3. **Plot the Y-Intercept**:
- Mark the point (0, 3) on the graph.
4. **Find Another Point Using the Slope**:
- Starting at (0, 3), move 2 units right and 1 unit down to reach (2, 2).
5. **Draw the Line**:
- Plot the points (0, 3) and (2, 2).
- Draw a straight line through these points, extending in both directions.
6. **Optional: Find the X-Intercept**:
- Set \( y = 0 \): \( x + 2(0) = 6 \) → \( x = 6 \), so the point is (6, 0).
- This confirms the line passes through (6, 0).
Now, you have successfully graphed the equation \( x + 2y = 6 \) using the slope and y-intercept.
Solution
Step 1: Write the equation in slope-intercept form.
Start with x + 2y = 6.
Subtract x from both sides to isolate the y-term:
2y = 6 − x
Divide both sides by 2:
y = 3 − (1/2)x
Step 2: Identify the slope and y-intercept.
The slope (m) is –1/2.
The y-intercept (b) is 3, which gives the point (0, 3).
Step 3: Plot the y-intercept.
Mark the point (0, 3) on your graph.
Step 4: Use the slope to find another point.
Since the slope is –1/2, for every 2 units you move to the right, move 1 unit down.
From (0, 3), moving 2 units right (to x = 2) and 1 unit down (to y = 2) gives the point (2, 2).
Step 5: Draw the line.
Plot the points (0, 3) and (2, 2). Draw a straight line through these points, extending in both directions.
Optional: Verify by finding the x-intercept.
Set y = 0 in the original equation:
x + 2(0) = 6 → x = 6, giving the point (6, 0).
This further confirms your plotted line.
You now have the graph of x + 2y = 6, drawn using the slope (–1/2) and the y-intercept (0, 3).
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Simplify this solution Extra Insights
To graph the equation \( x + 2y = 6 \), let's first rewrite it in slope-intercept form \( y = mx + b \). 1. Start with the original equation: \( 2y = -x + 6 \) 2. Divide everything by 2: \( y = -\frac{1}{2}x + 3 \) Now, we can see the slope \( m \) is \(-\frac{1}{2}\) and the \( y \)-intercept \( b \) is \( 3 \). To plot the graph, begin by marking the \( y \)-intercept (0, 3) on the graph. From there, use the slope to find another point: go down 1 unit (because of the negative slope) and right 2 units to reach (2, 2). Connect these points with a straight line, and you have your graph! It’s as simple as that!
