Write the equation of the line that is perpendicular to the given line described by the equation 6y=−12x−21
 and passes through the point (-4, 9).

We are asked to find the equation of a line that is perpendicular to the line \( 6y = -12x - 21 \) and passes through the point \( (-4, 9) \).

### Step 1: Convert the given equation to slope-intercept form

The equation \( 6y = -12x - 21 \) needs to be rewritten in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

First, divide both sides by 6 to solve for \( y \):
\[
y = \frac{-12}{6}x + \frac{-21}{6}
\]
\[
y = -2x - 3.5
\]
So, the slope of the given line is \( m = -2 \).

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

Lines that are perpendicular have slopes that are negative reciprocals of each other. If the slope of the given line is \( m = -2 \), then the slope of the perpendicular line, \( m_{\text{perpendicular}} \), is:
\[
m_{\text{perpendicular}} = \frac{1}{2}
\]

### Step 3: Use the point-slope form to find the equation

Now that we know the slope of the perpendicular line is \( \frac{1}{2} \), and it passes through the point \( (-4, 9) \), we can use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Where \( (x_1, y_1) = (-4, 9) \) and \( m = \frac{1}{2} \).

Substitute these values:
\[
y - 9 = \frac{1}{2}(x + 4)
\]

### Step 4: Simplify the equation

Distribute the slope on the right-hand side:
\[
y - 9 = \frac{1}{2}x + 2
\]
Now, add 9 to both sides to solve for \( y \):
\[
y = \frac{1}{2}x + 2 + 9
\]
\[
y = \frac{1}{2}x + 11
\]

### Final Equation:
The equation of the line that is perpendicular to \( 6y = -12x - 21 \) and passes through \( (-4, 9) \) is:
\[
y = \frac{1}{2}x + 11
\]

Would you like any additional details or have any questions? 

Here are 5 related questions for further practice:
1. What is the equation of a line perpendicular to \( y = 3x + 5 \) passing through \( (1, 2) \)?
2. How do you find the slope of a line parallel to \( 4y = 2x - 8 \)?
3. Write the equation of a line that is perpendicular to \( y = -\frac{3}{4}x + 1 \) and passes through \( (0, 0) \).
4. What is the distance between the points \( (1, 2) \) and \( (-4, 9) \)?
5. How do you convert an equation from standard form to slope-intercept form?

**Tip**: The negative reciprocal of a slope \( m \) is always \( -\frac{1}{m} \), which helps when finding perpendicular lines.

Write the equation of the line that is perpendicular to the given line described by the equation 6y = −12x − 21 and passes through the point (-4, 9).

Learn how to find the equation of a line that is perpendicular to the line 6y = −12x − 21 and passes through the point (-4, 9). This problem covers key algebraic concepts such as slope-intercept form, point-slope form, and the properties of perpendicular lines. Suitable for students in Grades 9-11.

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