Graph the equation $$ 5 x+6 y=-6 $$

Question

UpStudy AI · Accepted Answer

To graph the linear equation:

$$
5x + 6y = -6
$$

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 solve for $$ y $$:

$$
5x + 6y = -6
$$

Subtract $$5x$$ from both sides:

$$
6y = -5x - 6
$$

Divide every term by 6:

$$
y = -\frac{5}{6}x - 1
$$

**Slope ($$m$$)**: $$-\frac{5}{6}$$

**Y-intercept ($$b$$)**: $$-1$$

### 2. Find the Intercepts

**Y-intercept ($$b$$)**: This is the point where the line crosses the y-axis ($$x = 0$$).

$$
y = -1 \quad \Rightarrow \quad (0, -1)
$$

**X-intercept**: This is the point where the line crosses the x-axis ($$y = 0$$). Set $$ y = 0 $$ in the original equation and solve for $$ x $$:

$$
5x + 6(0) = -6 \
5x = -6 \
x = -\frac{6}{5} \quad \Rightarrow \quad \left(-\frac{6}{5}, 0\right)
$$

### 3. Plot the Points and Draw the Line

1. **Plot the Y-intercept**: (0, -1)
2. **Plot the X-intercept**: $$\left(-\frac{6}{5}, 0\right)$$ or $$(-1.2, 0)$$
3. **Draw the Line**: Connect these two points with a straight line extending in both directions.

### 4. Additional Points (Optional)

To ensure accuracy, you can plot additional points by choosing values for $$ x $$ or $$ y $$ and solving for the other variable.

**Example**:

Let $$ x = 6 $$:

$$
y = -\frac{5}{6}(6) - 1 = -5 - 1 = -6 \quad \Rightarrow \quad (6, -6)
$$

### 5. Graph Sketch

Here's a simple sketch of the graph:

```
y
|
|          
|         *
|        /
|       /
|      /
|     / 
|----*----------- x
       (-6/5,0)
```

- The point $$*$$ on the y-axis represents the y-intercept (0, -1).
- The point $$*$$ on the x-axis represents the x-intercept $$\left(-\frac{6}{5}, 0\right)$$.
- The line passes through both intercepts and extends infinitely in both directions.

### 6. Understanding the Graph

- **Slope ($$-\frac{5}{6}$$)**: For every 6 units you move to the right along the x-axis, the line moves 5 units downward.
- **Y-intercept ($$-1$$)**: The line crosses the y-axis at (0, -1).

### 7. Summary

The graph of the equation $$5x + 6y = -6$$ is a straight line with a slope of $$-\frac{5}{6}$$ that crosses the y-axis at $$(0, -1)$$ and the x-axis at $$\left(-\frac{6}{5}, 0\right)$$.
To graph the equation $$5x + 6y = -6$$, follow these steps: 1. **Slope-Intercept Form**: $$ y = -\frac{5}{6}x - 1 $$ - Slope ($$m$$) = $$-\frac{5}{6}$$ - Y-intercept ($$b$$) = $$-1$$ 2. **Intercepts**: - Y-intercept: (0, -1) - X-intercept: $$\left(-\frac{6}{5}, 0\right)$$ or (-1.2, 0) 3. **Plot Points**: - Plot (0, -1) and (-1.2, 0) on the graph. 4. **Draw the Line**: - Connect the two points with a straight line extending in both directions. 5. **Additional Points (Optional)**: - For example, when $$x = 6$$, $$y = -6$$, so plot (6, -6). 6. **Graph Summary**: - A straight line with a slope of $$-\frac{5}{6}$$ crossing the y-axis at (0, -1) and the x-axis at $$\left(-\frac{6}{5}, 0\right)$$. This graph represents all the points $$(x, y)$$ that satisfy the equation $$5x + 6y = -6$$.

Graph the equation \[ 5 x+6 y=-6 \]

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