Find the Equation Using Point-Slope Formula (-10,18) , (6,-14)

$(- 10, 18)$ , $(6, - 14)$

Step 1

Find the slope of the line between $(- 10, 18)$ and $(6, - 14)$ 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.

$m = \frac{- 14 - (18)}{6 - (- 10)}$

Simplify.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 14 - (18)$ and $6 - (- 10)$ .

Rewrite $- 14$ as $- 1 (14)$ .

$m = \frac{- 1 \cdot 14 - (18)}{6 - (- 10)}$

Factor $- 1$ out of $- 1 (14) - (18)$ .

$m = \frac{- 1 (14 + 18)}{6 - (- 10)}$

Reorder terms.

$m = \frac{- 1 (14 + 18)}{6 - 10 \cdot - 1}$

Factor $2$ out of $- 1 (14 + 18)$ .

$m = \frac{2 (- 1 (7 + 9))}{6 - 10 \cdot - 1}$

Cancel the common factors.

Factor $2$ out of $6$ .

$m = \frac{2 (- 1 (7 + 9))}{2 (3) - 10 \cdot - 1}$

Factor $2$ out of $- 10 \cdot - 1$ .

$m = \frac{2 (- 1 (7 + 9))}{2 (3) + 2 (- 5 \cdot - 1)}$

Factor $2$ out of $2 (3) + 2 (- 5 \cdot - 1)$ .

$m = \frac{2 (- 1 (7 + 9))}{2 (3 - 5 \cdot - 1)}$

Cancel the common factor.

$m = \frac{2 (- 1 (7 + 9))}{2 (3 - 5 \cdot - 1)}$

Rewrite the expression.

$m = \frac{- 1 (7 + 9)}{3 - 5 \cdot - 1}$

Add $7$ and $9$ .

$m = \frac{- 1 \cdot 16}{3 - 5 \cdot - 1}$

Simplify the denominator.

Multiply $- 5$ by $- 1$ .

$m = \frac{- 1 \cdot 16}{3 + 5}$

Add $3$ and $5$ .

$m = \frac{- 1 \cdot 16}{8}$

Simplify the expression.

Multiply $- 1$ by $16$ .

$m = \frac{- 16}{8}$

Divide $- 16$ by $8$ .

$m = - 2$

Step 2

Use the slope $- 2$ and a given point $(- 10, 18)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (18) = - 2 \cdot (x - (- 10))$

Step 3

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

$y - 18 = - 2 \cdot (x + 10)$

Step 4

Solve for $y$ .

Simplify $- 2 \cdot (x + 10)$ .

Rewrite.

$y - 18 = 0 + 0 - 2 \cdot (x + 10)$

Simplify by adding zeros.

$y - 18 = - 2 \cdot (x + 10)$

Apply the distributive property.

$y - 18 = - 2 x - 2 \cdot 10$

Multiply $- 2$ by $10$ .

$y - 18 = - 2 x - 20$

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

Add $18$ to both sides of the equation.

$y = - 2 x - 20 + 18$

Add $- 20$ and $18$ .

$y = - 2 x - 2$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 2 x - 2$

Point-slope form:

$y - 18 = - 2 \cdot (x + 10)$

Step 6

Find the Equation Using Point-Slope Formula (-10,18) , (6,-14)

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