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To solve this problem, we need to find the equation of the line that is **perpendicular** to the line \(2x - 3y = 12\) and passes through the point \((3, -2)\).

### Step 1: Find the slope of the given line
We start with the line equation:
\[
2x - 3y = 12
\]
To find the slope, we need to rewrite this in slope-intercept form (\(y = mx + b\)), where \(m\) is the slope.

\[
2x - 3y = 12
\]
Subtract \(2x\) from both sides:
\[
-3y = -2x + 12
\]
Now, divide by \(-3\) to isolate \(y\):
\[
y = \frac{2}{3}x - 4
\]
The slope of this line is \(\frac{2}{3}\).

### Step 2: Find the slope of the perpendicular line
The slope of the line perpendicular to a line is the negative reciprocal of the original slope. Since the slope of the original line is \(\frac{2}{3}\), the slope of the perpendicular line will be:
\[
-\frac{3}{2}
\]

### Step 3: Use the point-slope form of the equation
Now we use the point-slope form of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Here, \((x_1, y_1) = (3, -2)\) and \(m = -\frac{3}{2}\).

Substituting into the point-slope form:
\[
y - (-2) = -\frac{3}{2}(x - 3)
\]
Simplifying:
\[
y + 2 = -\frac{3}{2}(x - 3)
\]

### Step 4: Match with the options
Looking at the given options, this equation matches the third option:
\[
y + 2 = -\frac{3}{2}(x - 3)
\]

Thus, the correct answer is the third option: \(y + 2 = -\frac{3}{2}(x - 3)\).

Would you like further clarification or additional details?

Here are 5 related questions:
1. How do you find the slope of a line given two points?
2. What is the general formula for the slope-intercept form of a line?
3. How do you find the slope of a line perpendicular to a given line?
4. What is the difference between slope-intercept form and point-slope form?
5. How can you convert an equation from standard form to slope-intercept form?

**Tip:** Remember that perpendicular lines have slopes that are negative reciprocals of each other!

Which of the following is the equation of the line that is perpendicular to 2x - 3y = 12 and passes through the point (3, -2)?

Learn how to find the equation of a line that is perpendicular to 2x - 3y = 12 and passes through the point (3, -2). This problem covers key geometric concepts, including slopes of perpendicular lines and the point-slope form equation. Suitable for Grades 9-10 students.

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