Answer
The slope of the line is \(-\frac{1}{6}\) and the \( y \)-intercept is 6. The line crosses the \( y \)-axis at (0, 6) and slopes downward as it moves to the right.
Solution
To find the **slope** and **\( y \)-intercept** of the line given by the equation:
\[
x + 6y = 36
\]
follow these steps:
### 1. Rewrite the Equation in Slope-Intercept Form
The slope-intercept form of a linear equation is:
\[
y = mx + b
\]
where:
- \( m \) is the **slope**.
- \( b \) is the **\( y \)-intercept**.
Let's rearrange the given equation to match this form.
**Starting with:**
\[
x + 6y = 36
\]
**Subtract \( x \) from both sides:**
\[
6y = -x + 36
\]
**Divide every term by 6:**
\[
y = -\frac{1}{6}x + 6
\]
### 2. Identify the Slope and \( y \)-Intercept
From the equation \( y = -\frac{1}{6}x + 6 \):
- **Slope (\( m \))**: \( -\frac{1}{6} \)
- **\( y \)-Intercept (\( b \))**: \( 6 \)
This means the line crosses the \( y \)-axis at the point **(0, 6)** and has a slope of **\(-\frac{1}{6}\)**.
### 3. Graphing the Line
To graph the line \( y = -\frac{1}{6}x + 6 \), follow these steps:
1. **Plot the \( y \)-Intercept:**
- Start by plotting the point \( (0, 6) \) on the graph.
2. **Use the Slope to Find Another Point:**
- The slope \( -\frac{1}{6} \) means that for every **6 units** you move **to the right** on the \( x \)-axis, you move **1 unit down** on the \( y \)-axis.
- Starting from \( (0, 6) \):
- Move **6 units right** to \( x = 6 \).
- Move **1 unit down** to \( y = 5 \).
- Plot the point \( (6, 5) \).
3. **Draw the Line:**
- Connect the two points \( (0, 6) \) and \( (6, 5) \) with a straight line.
- Extend the line across the graph, ensuring it passes through both points.
4. **Additional Points (Optional):**
- To ensure accuracy, you can plot more points by choosing other values for \( x \) and solving for \( y \), or by finding the \( x \)-intercept.
### 4. Example Graph
Here's a simple representation of the graph:
```
y
|
8 |
7 |
6 |●
5 | ●
4 |
3 |
2 |
1 |
0 |________________ x
0 6 12 18 24 30 36
```
- **Point (0, 6):** Represented by the first ● on the \( y \)-axis.
- **Point (6, 5):** Represented by the second ●.
- The line connects these points and extends in both directions.
### 5. Summary
- **Slope (\( m \))**: \( -\frac{1}{6} \)
- **\( y \)-Intercept (\( b \))**: \( 6 \) (Point \( (0, 6) \))
- **Graph**: A straight line descending gently from left to right, crossing the \( y \)-axis at \( 6 \).
Feel free to plot additional points or use graphing software/tools for a more precise graph.
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
