To find the missing coordinates using the slope formula, we use the formula for the slope of a line:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Given that the slope $m = 7$, we'll use this formula to solve for the missing coordinates.

Problem 1: $(x, 8)$ and $(4, 1)$

The slope is 7, and we know the points $(x, 8)$ and $(4, 1)$. Using the slope formula:

$7 = \frac{1 - 8}{4 - x}$

This simplifies to:

$7 = \frac{-7}{4 - x}$

Now, solve for $x$:

$7(4 - x) = -7$

$28 - 7x = -7$

$28 + 7 = 7x$

$35 = 7x$

$x = 5$

So, the missing coordinate for the first pair is $x = 5$.

Problem 2: $(3, -6)$ and $(7, y)$

Again, the slope is 7, and we know the points $(3, -6)$ and $(7, y)$. Using the slope formula:

$7 = \frac{y - (-6)}{7 - 3}$

This simplifies to:

$7 = \frac{y + 6}{4}$

Now, solve for $y$:

$7 \times 4 = y + 6$

$28 = y + 6$

$y = 28 - 6$

$y = 22$

So, the missing coordinate for the second pair is $y = 22$.

Summary:

• For $(x, 8)$ and $(4, 1)$, $x = 5$
• For $(3, -6)$ and $(7, y)$, $y = 22$

Would you like further details on the solution or any clarifications? Here are 5 related questions for practice:

1. Find the slope between the points $(2, 3)$ and $(5, 15)$.
2. Given the points $(0, 0)$ and $(4, y)$, and a slope of 2, find $y$.
3. Find the equation of the line passing through $(2, 5)$ and $(4, 11)$.
4. If a line has a slope of 3 and passes through $(1, 4)$, find its equation in slope-intercept form.
5. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, what happens to the slope if $x_1 = x_2$?

Tip: Remember, when two points have the same x-coordinates, the slope is undefined because you cannot divide by zero!