Find the slope of a line perpendicular to the line whose equation is 3, x, minus, 6, y, equals, 1443*x*−6*y*=144. Fully simplify your answer.

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

### Step 1: Start with the given equation
The given equation is:

\[
3x - 6y = 144
\]

### Step 2: Solve for \( y \) in terms of \( x \)
First, isolate the \( y \)-term by subtracting \( 3x \) from both sides:

\[
-6y = -3x + 144
\]

Next, divide both sides by \( -6 \) to solve for \( y \):

\[
y = \frac{-3x + 144}{-6}
\]

Simplify each term:

\[
y = \frac{-3}{-6}x + \frac{144}{-6}
\]

\[
y = \frac{1}{2}x - 24
\]

### Step 3: Identify the slope of the original line
From the equation \( y = \frac{1}{2}x - 24 \), we can see that the slope \( m_1 \) of the original line is \( \frac{1}{2} \).

### Step 4: Find the slope of the perpendicular line
The slope of a line perpendicular to a given line is the negative reciprocal of the original slope. The negative reciprocal of \( \frac{1}{2} \) is \( -2 \).

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

\[
\boxed{-2}
\]

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Would you like further explanation on any part of this? 

Here are five related questions for you to consider:
1. How do you find the slope of a line parallel to a given line?
2. What is the geometric interpretation of the negative reciprocal of a slope?
3. How do you determine the equation of a line given its slope and a point on the line?
4. Can a line have no slope? What does it mean geometrically?
5. How do you write the equation of a line when given two points on the line?

**Tip**: When working with perpendicular lines, remember that their slopes are always negative reciprocals of each other.

Find the slope of a line perpendicular to the line whose equation is 3x - 6y = 144.

Learn how to find the slope of a line perpendicular to the equation 3x - 6y = 144 with a detailed step-by-step solution. This problem demonstrates the use of slope-intercept form and the concept of negative reciprocals for perpendicular lines. Suitable for students in Grades 8-10.

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