We are given the equation of line $p$:

$y = \frac{9}{7}x - 9$

Step 1: Find the Slope of Line $q$

Since line $q$ is perpendicular to line $p$, its slope is the negative reciprocal of the slope of $p$.

• The slope of line $p$ is $\frac{9}{7}$.
• The negative reciprocal of $\frac{9}{7}$ is $-\frac{7}{9}$.

Thus, the slope of line $q$ is $-\frac{7}{9}$.

Step 2: Use the Point-Slope Formula

Line $q$ passes through the point $(9, -6)$. Using the point-slope form:

$y - y_1 = m(x - x_1)$

where $(x_1, y_1) = (9, -6)$ and $m = -\frac{7}{9}$,

$y - (-6) = -\frac{7}{9} (x - 9)$

$y + 6 = -\frac{7}{9} (x - 9)$

Step 3: Expand and Solve for $y$

Distribute $-\frac{7}{9}$:

$y + 6 = -\frac{7}{9}x + 7$

Subtract 6 from both sides:

$y = -\frac{7}{9}x + 1$

Final Answer:

$y = -\frac{7}{9}x + 1$

Would you like a step-by-step explanation or more details on any part?

Related Questions:

1. How do you find the equation of a line given two points?
2. What is the general formula for perpendicular slopes?
3. How do you convert an equation from point-slope form to slope-intercept form?
4. Can you find the x- and y-intercepts of the equation $y = -\frac{7}{9}x + 1$?
5. What is the distance between the two given lines at any x-coordinate?

Tip:

The product of the slopes of two perpendicular lines is always -1.