Find the slope of a line perpendicular to the line whose equation is 3, x, minus, 6, y, equals, minus, 723*x*−6*y*=−72. Fully simplify your answer.

We are tasked with finding the slope of a line that is perpendicular to the line with the equation:

\[
3x - 6y = -72
\]

### Step 1: Rewrite the equation in slope-intercept form
To find the slope, we need to express the equation in the slope-intercept form, which is:

\[
y = mx + b
\]

Where \( m \) is the slope.

Start by solving for \( y \) in terms of \( x \):

\[
3x - 6y = -72
\]

Subtract \( 3x \) from both sides:

\[
-6y = -3x - 72
\]

Now, divide every term by \( -6 \) to isolate \( y \):

\[
y = \frac{-3x - 72}{-6}
\]

Simplify:

\[
y = \frac{-3x}{-6} + \frac{-72}{-6}
\]

\[
y = \frac{1}{2}x + 12
\]

Thus, the equation of the line in slope-intercept form is:

\[
y = \frac{1}{2}x + 12
\]

### Step 2: Identify the slope of the given line
From the equation \( y = \frac{1}{2}x + 12 \), the slope of the given line is \( \frac{1}{2} \).

### Step 3: Find the slope of the perpendicular line
The slopes of perpendicular lines are negative reciprocals of each other. If the slope of the original line is \( m_1 = \frac{1}{2} \), the slope of the perpendicular line, \( m_2 \), is given by:

\[
m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{1}{2}} = -2
\]

### Final Answer:
The slope of the line perpendicular to the given line is:

\[
\boxed{-2}
\]

Would you like a detailed breakdown of any part, or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the slope of a line parallel to the original line \( 3x - 6y = -72 \)?
2. How do you find the equation of a line passing through a point and perpendicular to another line?
3. What is the significance of the y-intercept in a line's equation?
4. How do you calculate the angle between two lines given their slopes?
5. How do perpendicular slopes behave in three-dimensional space?

**Tip:** When working with perpendicular lines, always remember that their slopes multiply to give \(-1\).

Find the slope of a line perpendicular to the line whose equation is 3x - 6y = -72. Fully simplify your answer.

Learn how to find the slope of a line perpendicular to the equation 3x - 6y = -72. This problem covers key algebraic concepts like linear equations, slope-intercept form, and perpendicular slopes. Suitable for students in Grades 8-10.

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