Find the Equation Using Point-Slope Formula (-12,8) , (9,1)

$(- 12, 8)$ , $(9, 1)$

Step 1

Find the slope of the line between $(- 12, 8)$ and $(9, 1)$ 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.

$m = \frac{1 - (8)}{9 - (- 12)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $8$ .

$m = \frac{1 - 8}{9 - (- 12)}$

Subtract $8$ from $1$ .

$m = \frac{- 7}{9 - (- 12)}$

Simplify the denominator.

Multiply $- 1$ by $- 12$ .

$m = \frac{- 7}{9 + 12}$

Add $9$ and $12$ .

$m = \frac{- 7}{21}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 7$ and $21$ .

Factor $7$ out of $- 7$ .

$m = \frac{7 (- 1)}{21}$

Cancel the common factors.

Factor $7$ out of $21$ .

$m = \frac{7 \cdot - 1}{7 \cdot 3}$

Cancel the common factor.

$m = \frac{7 \cdot - 1}{7 \cdot 3}$

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) , (9,1)

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