Graph this line using the slope and $$ y $$-intercept: $$ y=\frac{1}{9} x+9 $$ Click to select points on the graph.

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To graph the linear equation $$ y = \frac{1}{9}x + 9 $$ using its slope and $$ y $$-intercept, follow these steps:

### 1. Identify the $$ y $$-Intercept

The equation is in the slope-intercept form $$ y = mx + b $$, where:
- $$ m $$ is the slope.
- $$ b $$ is the $$ y $$-intercept.

For $$ y = \frac{1}{9}x + 9 $$:
- **Slope ($$ m $$)**: $$ \frac{1}{9} $$
- **$$ y $$-Intercept ($$ b $$)**: $$ 9 $$

**Plot the $$ y $$-Intercept:**
- Start by plotting the point $$ (0, 9) $$ on the graph. This is where the line crosses the $$ y $$-axis.

### 2. Use the Slope to Find Another Point

The slope $$ \frac{1}{9} $$ means that for every 9 units you move horizontally (to the right), you move up by 1 unit vertically.

**From the $$ y $$-Intercept $$ (0, 9) $$:**
- **Run (Horizontal Change)**: $$ +9 $$ units
- **Rise (Vertical Change)**: $$ +1 $$ unit

**Calculate the Next Point:**
- Move right 9 units from $$ x = 0 $$ to $$ x = 9 $$.
- Move up 1 unit from $$ y = 9 $$ to $$ y = 10 $$.
- **Second Point**: $$ (9, 10) $$

### 3. Plot Additional Points (Optional)

To ensure accuracy, you can plot more points using the slope:

- **Third Point**:
  - From $$ (9, 10) $$, move right another 9 units to $$ x = 18 $$.
  - Move up 1 unit to $$ y = 11 $$.
  - **Point**: $$ (18, 11) $$

- **Fourth Point**:
  - To the left of the $$ y $$-intercept, move left 9 units to $$ x = -9 $$.
  - Move down 1 unit to $$ y = 8 $$.
  - **Point**: $$ (-9, 8) $$

### 4. Draw the Line

- Using a ruler or straightedge, draw a straight line that passes through the plotted points.
- Extend the line in both directions and add arrowheads to indicate that it continues infinitely.

### 5. Summary of Key Points

| $$ x $$ | $$ y $$ |
|---------|---------|
| -9      | 8       |
| 0       | 9       |
| 9       | 10      |
| 18      | 11      |

### 6. Visual Representation

While I can't create an interactive graph here, here's a simple sketch to visualize:

```
 y
 ^
 |
11|                   • (18,11)
10|           • (9,10)
 9|• (0,9)
 8|       • (-9,8)
  |
  +-------------------------------> x
```

Each "•" represents a plotted point. Draw a straight line connecting these points to represent the equation $$ y = \frac{1}{9}x + 9 $$.

### 7. Verifying the Graph

To ensure accuracy, plug in the $$ x $$-values from your points back into the equation and verify that the corresponding $$ y $$-values satisfy the equation:
- For $$ x = 9 $$: $$ y = \frac{1}{9}(9) + 9 = 1 + 9 = 10 $$ ✅
- For $$ x = -9 $$: $$ y = \frac{1}{9}(-9) + 9 = -1 + 9 = 8 $$ ✅

This confirms that your graph accurately represents the equation.
To graph the line $$ y = \frac{1}{9}x + 9 $$, plot the $$ y $$-intercept at $$ (0, 9) $$. Then, use the slope $$ \frac{1}{9} $$ to find another point by moving 9 units right and 1 unit up to $$ (9, 10) $$. Draw a straight line through these points to complete the graph.

Graph this line using the slope and -intercept:

Click to select points on the graph.

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Graph this line using the slope and -intercept: Click to select points on the graph.