To find the equation of line $w$, which is perpendicular to line $v$ with the equation $y = -9x + 1$ and passes through the point $(3, -2)$, let's go through the following steps:

Step 1: Find the Slope of Line $v$

The equation of line $v$ is given in slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

For $v$: $m = -9$

Step 2: Find the Slope of Line $w$ (Perpendicular Slope)

Since line $w$ is perpendicular to line $v$, the slope of $w$ will be the negative reciprocal of the slope of $v$.

Thus, the slope of $w$ is: $m_w = \frac{1}{9}$

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

Line $w$ passes through the point $(3, -2)$, and its slope is $\frac{1}{9}$. We can use the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$ where $(x_1, y_1) = (3, -2)$ and $m = \frac{1}{9}$.

Substitute these values: $y - (-2) = \frac{1}{9}(x - 3)$ $y + 2 = \frac{1}{9}x - \frac{1}{3}$

Step 4: Simplify to Slope-Intercept Form

Now, isolate $y$ to put it in slope-intercept form ($y = mx + b$):

$y = \frac{1}{9}x - \frac{1}{3} - 2$

Convert $-2$ to a fraction with a denominator of 3: $y = \frac{1}{9}x - \frac{1}{3} - \frac{6}{3}$ $y = \frac{1}{9}x - \frac{7}{3}$

Final Answer

The equation of line $w$ in slope-intercept form is: $y = \frac{1}{9}x - \frac{7}{3}$

Would you like further details on any of these steps or have additional questions?

Here are five related questions to deepen your understanding:

1. How would the equation change if line $w$ passed through a different point?
2. What if line $w$ was parallel to line $v$ instead of perpendicular?
3. Can we derive the equation of a perpendicular line in other forms besides slope-intercept?
4. How can we verify if two lines are perpendicular?
5. What is the significance of the y-intercept in interpreting the graph of line $w$?

Tip: Always remember that perpendicular lines have slopes that are negative reciprocals of each other!