To solve this problem, let’s go through it step by step:

Step 1: Find the Slope of the Given Line

The line passes through the points $(-2, 1)$ and $(6, 7)$.

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the given points: $m = \frac{7 - 1}{6 - (-2)} = \frac{6}{8} = \frac{3}{4}$

Step 2: Find the Slope of the Perpendicular Line

Since we need a line that is perpendicular to this one, we take the negative reciprocal of the slope. The perpendicular slope, $m_{\perp}$, is: $m_{\perp} = -\frac{4}{3}$

Step 3: Use Point-Slope Form to Find the Equation of the Line

We want the equation of the line that passes through the point $(-6, 9)$ with slope $-\frac{4}{3}$.

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Substitute $m = -\frac{4}{3}$ and the point $(-6, 9)$: $y - 9 = -\frac{4}{3}(x + 6)$

Step 4: Simplify to Slope-Intercept Form

Expand and simplify to get $y$ in terms of $x$: $y - 9 = -\frac{4}{3}x - 8$ $y = -\frac{4}{3}x + 1$

Final Answer

The equation of the line in slope-intercept form is: $y = -\frac{4}{3}x + 1$

Would you like a detailed explanation on any of the steps?

Related Questions:

1. How do you determine the slope between any two points?
2. Why is the slope of a perpendicular line the negative reciprocal?
3. What are the steps to convert from point-slope form to slope-intercept form?
4. How do you verify if two lines are perpendicular?
5. What are some practical applications of perpendicular lines in geometry?

Tip:

Remember, for perpendicular lines, the product of their slopes is always $-1$.