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.