Graph these equations: $$ 2 x-3 y=-12 $$ $$ 2 x-3 y=-12 $$

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UpStudy AI · Accepted Answer

It looks like you've provided two identical linear equations:

$$
2x - 3y = -12
$$
$$
2x - 3y = -12
$$

Since both equations are the same, their graphs will coincide, representing a single straight line on the Cartesian plane. Here's how you can understand and visualize this line:

### 1. **Convert to Slope-Intercept Form**

First, let's rewrite the equation in the slope-intercept form, $$ y = mx + b $$, where $$ m $$ is the slope and $$ b $$ is the y-intercept.

$$
2x - 3y = -12
$$

**Step-by-Step Conversion:**

1. **Isolate the term with $$ y $$:**
   
   $$
   -3y = -2x - 12
   $$

2. **Solve for $$ y $$ by dividing all terms by -3:**
   
   $$
   y = \frac{2}{3}x + 4
   $$

So, the slope-intercept form is:

$$
y = \frac{2}{3}x + 4
$$

### 2. **Identify Key Features**

- **Slope ($$ m $$)**: $$ \frac{2}{3} $$
  
  This means that for every 3 units you move horizontally to the right, the line moves up by 2 units.

- **Y-intercept ($$ b $$)**: $$ 4 $$
  
  The line crosses the y-axis at the point $$ (0, 4) $$.

### 3. **Plotting the Line**

To graph the line, you can plot the y-intercept and use the slope to find another point.

1. **Plot the Y-Intercept:**
   
   - Point: $$ (0, 4) $$

2. **Use the Slope to Find Another Point:**
   
   - From $$ (0, 4) $$, move right 3 units (along the x-axis) and up 2 units (along the y-axis) to reach the point $$ (3, 6) $$.

3. **Draw the Line:**
   
   - Connect these two points with a straight line extending in both directions.

### 4. **Visual Representation**

While I can't provide a visual graph here, I can describe it:

- The line crosses the y-axis at $$ (0, 4) $$.
- It passes through $$ (3, 6) $$.
- It continues infinitely in both directions following the slope $$ \frac{2}{3} $$.

### 5. **Summary**

- **Equation:** $$ y = \frac{2}{3}x + 4 $$
- **Slope:** $$ \frac{2}{3} $$
- **Y-intercept:** $$ (0, 4) $$
- **Graph:** A straight line passing through $$ (0, 4) $$ and $$ (3, 6) $$, extending infinitely in both directions.

Since both equations you've provided are identical, their graphs are the same line. If you intended to graph two different equations, please double-check and provide the correct second equation!
Both equations represent the same straight line $$ y = \frac{2}{3}x + 4 $$.

Graph these equations:

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