Through (3, −9);  perpendicular to the line passing through the points (−4, 6) and (5, 3)

To solve this problem, we need to find the equation of a line that passes through the point (3, -9) and is perpendicular to the line passing through the points (-4, 6) and (5, 3).

### Step 1: Find the slope of the given line

We start by calculating the slope \( m_1 \) of the line passing through the points (-4, 6) and (5, 3). The slope formula is:

\[
m_1 = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the coordinates:

\[
m_1 = \frac{3 - 6}{5 - (-4)} = \frac{-3}{9} = -\frac{1}{3}
\]

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

The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Since the slope of the original line is \( -\frac{1}{3} \), the slope \( m_2 \) of the perpendicular line is:

\[
m_2 = 3
\]

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

We now use the point-slope form of the equation of a line, which is:

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

Here, \( m = 3 \) (the slope of the perpendicular line), and \( (x_1, y_1) = (3, -9) \). Substituting these values into the equation:

\[
y - (-9) = 3(x - 3)
\]

Simplifying:

\[
y + 9 = 3(x - 3)
\]
\[
y + 9 = 3x - 9
\]
\[
y = 3x - 9 - 9
\]
\[
y = 3x - 18
\]

### Final Equation

The equation of the line that passes through the point (3, -9) and is perpendicular to the line passing through (-4, 6) and (5, 3) is:

\[
y = 3x - 18
\]

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Would you like a deeper explanation or further details on any step?

Here are 5 related questions to expand on this:

1. How do you find the slope of a line given two points?
2. What is the significance of the negative reciprocal in perpendicular lines?
3. How can we convert the point-slope form to slope-intercept form?
4. What are other methods to check if two lines are perpendicular?
5. Can we generalize this method for lines in 3D space?

**Tip:** The slopes of two perpendicular lines in 2D space multiply to give -1.

Through (3, −9); perpendicular to the line passing through the points (−4, 6) and (5, 3)

Learn how to find the equation of a line that passes through the point (3, −9) and is perpendicular to the line passing through the points (−4, 6) and (5, 3). This step-by-step solution involves calculating the slope, finding the negative reciprocal for perpendicularity, and applying the point-slope formula. Suitable for students in Grades 9-12.

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