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 find the equation of the line in slope-intercept form that passes through the point \((-6, 9)\) and is perpendicular to a line passing through \((-2, 1)\) and \((6, 7)\), we can follow these steps:

### Step 1: Find the slope of the original line
The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-2, 1)\) and \((6, 7)\), we have:
\[
m = \frac{7 - 1}{6 - (-2)} = \frac{6}{8} = \frac{3}{4}
\]
So, the slope of the line passing through \((-2, 1)\) and \((6, 7)\) is \( \frac{3}{4} \).

### Step 2: Find the perpendicular slope
Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of the line perpendicular to \( \frac{3}{4} \) is:
\[
m_{\text{perpendicular}} = -\frac{4}{3}
\]

### Step 3: Use the point-slope form
Now, we have a slope of \( -\frac{4}{3} \) and a point \((-6, 9)\) through which the line passes. We can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -\frac{4}{3} \), \( x_1 = -6 \), and \( y_1 = 9 \):
\[
y - 9 = -\frac{4}{3}(x + 6)
\]

### Step 4: Simplify to slope-intercept form
Distribute \( -\frac{4}{3} \):
\[
y - 9 = -\frac{4}{3}x - 8
\]
Add 9 to both sides to solve for \( y \):
\[
y = -\frac{4}{3}x + 1
\]

### Final Answer
The equation of the line in slope-intercept form is:
\[
y = -\frac{4}{3}x + 1
\]

Would you like more details or have any questions?

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Here are 5 related questions for practice:

1. How do you find the equation of a line passing through two points?
2. What is the significance of perpendicular slopes being negative reciprocals?
3. How would the equation change if the line passed through a different point, say \((2, -3)\)?
4. How do you convert from point-slope form to slope-intercept form?
5. If two lines are parallel, how would their slopes compare?

**Tip:** When working with slopes, always double-check if you need a positive or negative reciprocal, especially when dealing with perpendicular lines!

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 write the equation of a line in slope-intercept form that passes through the point (−6, 9) and is perpendicular to a line passing through (−2, 1) and (6, 7). This problem covers slope calculations, perpendicular lines, and linear equations. Suitable for Grades 8-10.

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