We are given the equation of line $u$ in point-slope form as:

$y - 3 = 7(x - 5)$

Step 1: Convert the equation of line $u$ to slope-intercept form

To do this, we'll expand the equation:

$y - 3 = 7(x - 5)$

Distribute the 7 on the right side:

$y - 3 = 7x - 35$

Now, add 3 to both sides to isolate $y$:

$y = 7x - 35 + 3$

$y = 7x - 32$

Thus, the equation of line $u$ is:

$y = 7x - 32$

Step 2: Identify the slope of line $u$

From the equation $y = 7x - 32$, we see that the slope of line $u$ is $m = 7$.

Step 3: Determine the equation of line $v$

Since line $v$ is parallel to line $u$, it will have the same slope, $m = 7$.

We are also told that line $v$ passes through the point $(-1, -9)$. Now, we can use the point-slope form to write the equation of line $v$:

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

Substitute $m = 7$, $x_1 = -1$, and $y_1 = -9$:

$y - (-9) = 7(x - (-1))$

Simplify:

$y + 9 = 7(x + 1)$

Step 4: Convert to slope-intercept form

Expand the equation:

$y + 9 = 7x + 7$

Subtract 9 from both sides:

$y = 7x + 7 - 9$

$y = 7x - 2$

Final Answer:

The equation of line $v$ in slope-intercept form is:

$y = 7x - 2$

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Would you like more details or have any questions?

Here are some related questions for further practice:

1. What is the equation of a line perpendicular to $u$ passing through the point $(-1, -9)$?
2. What is the equation of a line parallel to $u$ passing through $(2, 3)$?
3. How do you find the distance between two parallel lines?
4. What is the slope of a line perpendicular to line $v$?
5. How would you graph the equation of line $v$?

Tip: When two lines are parallel, they always have the same slope!