4. Which equation represents a line perpendicular to the line whose equation is $$ 2 x+3 y=12 $$ ? $$ \begin{array}{ll} ext { (1) } 6 y=-4 x+12 & ext { (3) } 2 y=-3 x+6 \ ext { (2) } 2 y=3 x+6 & ext { (4) } 3 y=-2 x+12\end{array} $$

Question

4. Which equation represents a line perpendicular to the line whose equation is $$ 2 x+3 y=12 $$ ? $$ \begin{array}{ll}	ext { (1) } 6 y=-4 x+12 & 	ext { (3) } 2 y=-3 x+6 \ 	ext { (2) } 2 y=3 x+6 & 	ext { (4) } 3 y=-2 x+12\end{array} $$

UpStudy AI · Accepted Answer

To find the equation of a line that is perpendicular to the line given by the equation $$ 2x + 3y = 12 $$, we need to follow these steps:

1. **Convert the given equation to slope-intercept form (y = mx + b)** to find its slope.
2. **Determine the slope of the perpendicular line**. The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
3. **Check each of the provided options** to see which one has the correct slope.

### Step 1: Convert to slope-intercept form

Starting with the equation:
$$
2x + 3y = 12
$$

We can solve for $$ y $$:
$$
3y = -2x + 12
$$
$$
y = -\frac{2}{3}x + 4
$$

From this, we see that the slope $$ m $$ of the original line is $$ -\frac{2}{3} $$.

### Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to this one is the negative reciprocal of $$ -\frac{2}{3} $$:
$$
m_{\text{perpendicular}} = \frac{3}{2}
$$

### Step 3: Check the options

Now we will check each of the provided options to see which one has a slope of $$ \frac{3}{2} $$.

1. **Option (1)**: $$ 6y = -4x + 12 $$
   - Convert to slope-intercept form:
   $$
   y = -\frac{4}{6}x + 2 = -\frac{2}{3}x + 2 \quad \text{(slope = } -\frac{2}{3}\text{)}
   $$

2. **Option (2)**: $$ 2y = 3x + 6 $$
   - Convert to slope-intercept form:
   $$
   y = \frac{3}{2}x + 3 \quad \text{(slope = } \frac{3}{2}\text{)}
   $$

3. **Option (3)**: $$ 2y = -3x + 6 $$
   - Convert to slope-intercept form:
   $$
   y = -\frac{3}{2}x + 3 \quad \text{(slope = } -\frac{3}{2}\text{)}
   $$

4. **Option (4)**: $$ 3y = -2x + 12 $$
   - Convert to slope-intercept form:
   $$
   y = -\frac{2}{3}x + 4 \quad \text{(slope = } -\frac{2}{3}\text{)}
   $$

### Conclusion

The only option that has a slope of $$ \frac{3}{2} $$ is **Option (2)**: $$ 2y = 3x + 6 $$.

Thus, the equation that represents a line perpendicular to the line $$ 2x + 3y = 12 $$ is:
$$
\boxed{2y = 3x + 6}
$$
The equation $$ 2y = 3x + 6 $$ is perpendicular to $$ 2x + 3y = 12 $$.

Which equation represents a line perpendicular to the line whose equation is ?

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