Find the Equation Using Point-Slope Formula (-12,8) , (6,2)

$(- 12, 8)$ , $(6, 2)$

Step 1

Find the slope of the line between $(- 12, 8)$ and $(6, 2)$ 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.

$m = \frac{2 - (8)}{6 - (- 12)}$

Simplify.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2 - (8)$ and $6 - (- 12)$ .

Rewrite $2$ as $- 1 (- 2)$ .

$m = \frac{- 1 \cdot - 2 - (8)}{6 - (- 12)}$

Factor $- 1$ out of $- 1 (- 2) - (8)$ .

$m = \frac{- 1 (- 2 + 8)}{6 - (- 12)}$

Reorder terms.

$m = \frac{- 1 (- 2 + 8)}{6 - 12 \cdot - 1}$

Factor $2$ out of $- 1 (- 2 + 8)$ .

$m = \frac{2 (- 1 (- 1 + 4))}{6 - 12 \cdot - 1}$

Cancel the common factors.

Factor $2$ out of $6$ .

$m = \frac{2 (- 1 (- 1 + 4))}{2 (3) - 12 \cdot - 1}$

Factor $2$ out of $- 12 \cdot - 1$ .

$m = \frac{2 (- 1 (- 1 + 4))}{2 (3) + 2 (- 6 \cdot - 1)}$

Factor $2$ out of $2 (3) + 2 (- 6 \cdot - 1)$ .

$m = \frac{2 (- 1 (- 1 + 4))}{2 (3 - 6 \cdot - 1)}$

Cancel the common factor.

$m = \frac{2 (- 1 (- 1 + 4))}{2 (3 - 6 \cdot - 1)}$

Rewrite the expression.

$m = \frac{- 1 (- 1 + 4)}{3 - 6 \cdot - 1}$

Add $- 1$ and $4$ .

$m = \frac{- 1 \cdot 3}{3 - 6 \cdot - 1}$

Simplify the denominator.

Multiply $- 6$ by $- 1$ .

$m = \frac{- 1 \cdot 3}{3 + 6}$

Add $3$ and $6$ .

$m = \frac{- 1 \cdot 3}{9}$

Reduce the expression by cancelling the common factors.

Multiply $- 1$ by $3$ .

$m = \frac{- 3}{9}$

Cancel the common factor of $- 3$ and $9$ .

Factor $3$ out of $- 3$ .

$m = \frac{3 (- 1)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

$m = \frac{- 1}{3}$

Move the negative in front of the fraction.

$m = - \frac{1}{3}$

Step 2

Use the slope $- \frac{1}{3}$ and a given point $(- 12, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = - \frac{1}{3} \cdot (x - (- 12))$

Step 3

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

$y - 8 = - \frac{1}{3} \cdot (x + 12)$

Step 4

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x + 12)$ .

Rewrite.

$y - 8 = 0 + 0 - \frac{1}{3} \cdot (x + 12)$

Simplify by adding zeros.

$y - 8 = - \frac{1}{3} \cdot (x + 12)$

Apply the distributive property.

$y - 8 = - \frac{1}{3} x - \frac{1}{3} \cdot 12$

Combine $x$ and $\frac{1}{3}$ .

$y - 8 = - \frac{x}{3} - \frac{1}{3} \cdot 12$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y - 8 = - \frac{x}{3} + \frac{- 1}{3} \cdot 12$

Factor $3$ out of $12$ .

$y - 8 = - \frac{x}{3} + \frac{- 1}{3} \cdot (3 (4))$

Cancel the common factor.

$y - 8 = - \frac{x}{3} + \frac{- 1}{3} \cdot (3 \cdot 4)$

Rewrite the expression.

$y - 8 = - \frac{x}{3} - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$y - 8 = - \frac{x}{3} - 4$

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

Add $8$ to both sides of the equation.

$y = - \frac{x}{3} - 4 + 8$

Add $- 4$ and $8$ .

$y = - \frac{x}{3} + 4$

Step 5

Reorder terms.

$y = - (\frac{1}{3} x) + 4$

Step 6

Remove parentheses.

$y = - \frac{1}{3} x + 4$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{1}{3} x + 4$

Point-slope form:

$y - 8 = - \frac{1}{3} \cdot (x + 12)$

Step 8

Find the Equation Using Point-Slope Formula (-12,8) , (6,2)

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