Find the slope of a line perpendicular to the line whose equation is 3, x, minus, 2, y, equals, 63x−2y=6. Fully simplify your answer.

We are given the equation of a line:

\[
3x - 2y = 6
\]

To find the slope of a line perpendicular to this one, we first need to find the slope of the given line. We do this by rewriting the equation in **slope-intercept form** \(y = mx + b\), where \(m\) represents the slope.

### Step 1: Solve for \(y\)
Start with the equation:
\[
3x - 2y = 6
\]
Subtract \(3x\) from both sides:
\[
-2y = -3x + 6
\]
Now, divide both sides by \(-2\) to isolate \(y\):
\[
y = \frac{-3x + 6}{-2}
\]
Simplify each term:
\[
y = \frac{3}{2}x - 3
\]

So, the slope of the given line is:
\[
m = \frac{3}{2}
\]

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \(\frac{3}{2}\) is:

\[
-\frac{2}{3}
\]

### Final Answer:
The slope of the line perpendicular to the given line is:
\[
-\frac{2}{3}
\]

Would you like to go over the steps again or have any further questions?

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Here are 5 related questions:
1. How do you find the slope of a line given two points?
2. What is the slope of a line parallel to the given line?
3. How would you write the equation of a line with a given slope and point?
4. What is the significance of perpendicular slopes in geometry?
5. Can two lines have the same slope but not be parallel?

**Tip:** The slope of a horizontal line is always 0, while the slope of a vertical line is undefined.

Find the slope of a line perpendicular to the line whose equation is 3x−2y=6. Fully simplify your answer.

Learn how to find the slope of a line perpendicular to the equation 3x−2y=6. This step-by-step guide explores the slope-intercept form and the rule for perpendicular slopes. Ideal for students in Grades 8-10.

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