Write the equation of a line that passes through the point (− 6, 9) and is perpendicular to a line that passes through the points (− 2, 1) and (6, 7) in slope-intercept form.

To write the equation of a line that passes through the point \((-6, 9)\) and is perpendicular to a line that passes through the points \((-2, 1)\) and \((6, 7)\), we can follow these steps:

### Step 1: Find the slope of the line passing through points \((-2, 1)\) and \((6, 7)\)
The slope \(m_1\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

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

Substituting the given points \((-2, 1)\) and \((6, 7)\):

\[
m_1 = \frac{7 - 1}{6 - (-2)} = \frac{6}{6 + 2} = \frac{6}{8} = \frac{3}{4}
\]

### Step 2: Find the slope of the perpendicular line
The slope of the line perpendicular to the one found above will be the negative reciprocal of \(m_1\). If the slope of one line is \(m_1\), the slope of the perpendicular line \(m_2\) is:

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

### Step 3: Write the equation of the perpendicular line
Now we know the slope \(m_2 = -\frac{4}{3}\) and the line passes through the point \((-6, 9)\). The equation of a line in slope-intercept form is given by:

\[
y = mx + b
\]

We can substitute the slope \(m = -\frac{4}{3}\) and the point \((-6, 9)\) into this equation to solve for \(b\), the y-intercept.

Substitute \(x = -6\), \(y = 9\), and \(m = -\frac{4}{3}\):

\[
9 = -\frac{4}{3}(-6) + b
\]

Simplify:

\[
9 = 8 + b
\]

Solving for \(b\):

\[
b = 9 - 8 = 1
\]

### Step 4: Final equation
The equation of the line is:

\[
y = -\frac{4}{3}x + 1
\]

This is the equation of the line in slope-intercept form that passes through the point \((-6, 9)\) and is perpendicular to the line passing through the points \((-2, 1)\) and \((6, 7)\).

Do you need any further explanation or details?

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Here are 5 related questions to deepen your understanding:
1. How do you find the equation of a line that is parallel to a given line?
2. What is the geometric interpretation of a negative slope?
3. Can a vertical line be perpendicular to a horizontal line? Why?
4. How do you find the midpoint between two points on a line?
5. What is the significance of the slope-intercept form of a line in graphing?

### Tip: 
When solving for the slope of a line, always be careful with the signs when subtracting coordinates to avoid common errors.

Write the equation of a line that passes through the point (−6, 9) and is perpendicular to a line that passes through the points (−2, 1) and (6, 7) in slope-intercept form.

Learn how to find the equation of a line passing through the point (-6, 9) that is perpendicular to a line through the points (-2, 1) and (6, 7). This problem uses algebraic concepts such as slope, perpendicular slopes, and slope-intercept form, suitable for students in Grades 8-10.

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