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The equation provided is:

\[
3x - 6y = 90
\]

To find the slope of a line perpendicular to this one, we first need to convert the given equation into slope-intercept form, \( y = mx + b \), where \( m \) is the slope.

### Step 1: Rearrange the equation into slope-intercept form.

Start with:
\[
3x - 6y = 90
\]

Solve for \( y \):
\[
-6y = -3x + 90
\]

Divide through by \(-6\) to isolate \( y \):
\[
y = \frac{1}{2}x - 15
\]

So, the slope of the given line is \( \frac{1}{2} \).

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

The slopes of two perpendicular lines are negative reciprocals of each other. The negative reciprocal of \( \frac{1}{2} \) is \( -2 \).

Thus, the slope of the line perpendicular to the given line is:
\[
m_{\text{perpendicular}} = -2
\]

Would you like more details on any step or have any questions?

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Here are some related questions you might find interesting:
1. What is the slope of a parallel line to the equation \( 3x - 6y = 90 \)?
2. How do you find the y-intercept of the original line \( 3x - 6y = 90 \)?
3. What does the negative reciprocal of a slope represent geometrically?
4. Can two lines with the same slope ever be perpendicular?
5. How does changing the constant in a linear equation affect the slope?

**Tip**: When converting a linear equation to slope-intercept form, always isolate \( y \) by carefully rearranging terms and dividing coefficients.

Find the slope of a line perpendicular to the line whose equation is 3x - 6y = 90. Fully simplify your answer.

Learn how to find the slope of a line perpendicular to the given equation 3x - 6y = 90. This problem involves converting the equation into slope-intercept form and understanding the concept of negative reciprocals of slopes. Suitable for students in Grades 8-10.

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