Find the Equation Using Point-Slope Formula (8,10) , (-4,2)

$(8, 10)$ , $(- 4, 2)$

Step 1

Find the slope of the line between $(8, 10)$ and $(- 4, 2)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{2 - (10)}{- 4 - (8)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $10$ .

$m = \frac{2 - 10}{- 4 - (8)}$

Subtract $10$ from $2$ .

$m = \frac{- 8}{- 4 - (8)}$

Simplify the denominator.

Multiply $- 1$ by $8$ .

$m = \frac{- 8}{- 4 - 8}$

Subtract $8$ from $- 4$ .

$m = \frac{- 8}{- 12}$

Cancel the common factor of $- 8$ and $- 12$ .

Factor $- 4$ out of $- 8$ .

$m = \frac{- 4 \cdot 2}{- 12}$

Cancel the common factors.

Factor $- 4$ out of $- 12$ .

$m = \frac{- 4 \cdot 2}{- 4 \cdot 3}$

Cancel the common factor.

$m = \frac{- 4 \cdot 2}{- 4 \cdot 3}$

Rewrite the expression.

$m = \frac{2}{3}$

Step 2

Use the slope $\frac{2}{3}$ and a given point $(8, 10)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (10) = \frac{2}{3} \cdot (x - (8))$

Step 3

Simplify the equation and keep it in point-slope form.

$y - 10 = \frac{2}{3} \cdot (x - 8)$

Step 4

Solve for $y$ .

Simplify $\frac{2}{3} \cdot (x - 8)$ .

Rewrite.

$y - 10 = 0 + 0 + \frac{2}{3} \cdot (x - 8)$

Simplify by adding zeros.

$y - 10 = \frac{2}{3} \cdot (x - 8)$

Apply the distributive property.

$y - 10 = \frac{2}{3} x + \frac{2}{3} \cdot - 8$

Combine $\frac{2}{3}$ and $x$ .

$y - 10 = \frac{2 x}{3} + \frac{2}{3} \cdot - 8$

Multiply $\frac{2}{3} \cdot - 8$ .

Combine $\frac{2}{3}$ and $- 8$ .

$y - 10 = \frac{2 x}{3} + \frac{2 \cdot - 8}{3}$

Multiply $2$ by $- 8$ .

$y - 10 = \frac{2 x}{3} + \frac{- 16}{3}$

Move the negative in front of the fraction.

$y - 10 = \frac{2 x}{3} - \frac{16}{3}$

Move all terms not containing $y$ to the right side of the equation.

Add $10$ to both sides of the equation.

$y = \frac{2 x}{3} - \frac{16}{3} + 10$

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{2 x}{3} - \frac{16}{3} + 10 \cdot \frac{3}{3}$

Combine $10$ and $\frac{3}{3}$ .

$y = \frac{2 x}{3} - \frac{16}{3} + \frac{10 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{2 x}{3} + \frac{- 16 + 10 \cdot 3}{3}$

Simplify the numerator.

Multiply $10$ by $3$ .

$y = \frac{2 x}{3} + \frac{- 16 + 30}{3}$

Add $- 16$ and $30$ .

$y = \frac{2 x}{3} + \frac{14}{3}$

Step 5

Reorder terms.

$y = \frac{2}{3} x + \frac{14}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{2}{3} x + \frac{14}{3}$

Point-slope form:

$y - 10 = \frac{2}{3} \cdot (x - 8)$

Step 7

Find the Equation Using Point-Slope Formula (8,10) , (-4,2)

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