write the equation of the line that passes through (-18, 11) and perpendicular to -3x+4y=9

To find the equation of the line that passes through the point \((-18, 11)\) and is perpendicular to the line given by \(-3x + 4y = 9\), we can follow these steps:

### Step 1: Find the slope of the given line.
The equation of the line is \(-3x + 4y = 9\). We first need to write it in slope-intercept form \(y = mx + b\), where \(m\) is the slope.

\[
4y = 3x + 9
\]

\[
y = \frac{3}{4}x + \frac{9}{4}
\]

So, the slope \(m_1\) of the given line is \(\frac{3}{4}\).

### Step 2: Determine the slope of the perpendicular line.
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope \(m_2\) of the perpendicular line is:

\[
m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{3}{4}} = -\frac{4}{3}
\]

### Step 3: Use the point-slope form to find the equation of the line.
The point-slope form of a line is given by:

\[
y - y_1 = m(x - x_1)
\]

Substituting the point \((-18, 11)\) and the slope \(-\frac{4}{3}\):

\[
y - 11 = -\frac{4}{3}(x + 18)
\]

### Step 4: Simplify the equation.
Distribute the slope on the right side:

\[
y - 11 = -\frac{4}{3}x - 24
\]

Add 11 to both sides to solve for \(y\):

\[
y = -\frac{4}{3}x - 13
\]

### Final Answer:
The equation of the line that passes through \((-18, 11)\) and is perpendicular to \(-3x + 4y = 9\) is:

\[
\boxed{y = -\frac{4}{3}x - 13}
\]

Would you like further details on any of these steps? Here are some related questions:

1. How do you find the slope of a line given two points?
2. What is the geometric significance of the negative reciprocal in finding perpendicular lines?
3. How do you convert an equation from standard form to slope-intercept form?
4. How can you graph a line given its equation in slope-intercept form?
5. What are the steps to find the distance between a point and a line?

**Tip:** When given a problem involving perpendicular lines, remember that their slopes are negative reciprocals of each other.

Learn how to find the equation of a line that is perpendicular to the line given by -3x + 4y = 9 and passes through the point (-18, 11). This problem involves understanding slopes, point-slope form, and the concept of perpendicular lines in algebra. Suitable for high school students.

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