Find the Equation Using Point-Slope Formula (9,-2) and (-3,2)

$(9, - 2)$ and $(- 3, 2)$

Step 1

Find the slope of the line between $(9, - 2)$ and $(- 3, 2)$ 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.

$m = \frac{2 - (- 2)}{- 3 - (9)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{2 + 2}{- 3 - (9)}$

Add $2$ and $2$ .

$m = \frac{4}{- 3 - (9)}$

Simplify the denominator.

Multiply $- 1$ by $9$ .

$m = \frac{4}{- 3 - 9}$

Subtract $9$ from $- 3$ .

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $4$ and $- 12$ .

Factor $4$ out of $4$ .

$m = \frac{4 (1)}{- 12}$

Cancel the common factors.

Factor $4$ out of $- 12$ .

$m = \frac{4 \cdot 1}{4 \cdot - 3}$

Cancel the common factor.

$m = \frac{4 \cdot 1}{4 \cdot - 3}$

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 $(9, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = - \frac{1}{3} \cdot (x - (9))$

Step 3

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

$y + 2 = - \frac{1}{3} \cdot (x - 9)$

Step 4

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x - 9)$ .

Rewrite.

$y + 2 = 0 + 0 - \frac{1}{3} \cdot (x - 9)$

Simplify by adding zeros.

$y + 2 = - \frac{1}{3} \cdot (x - 9)$

Apply the distributive property.

$y + 2 = - \frac{1}{3} x - \frac{1}{3} \cdot - 9$

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

$y + 2 = - \frac{x}{3} - \frac{1}{3} \cdot - 9$

Cancel the common factor of $3$ .

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

$y + 2 = - \frac{x}{3} + \frac{- 1}{3} \cdot - 9$

Factor $3$ out of $- 9$ .

$y + 2 = - \frac{x}{3} + \frac{- 1}{3} \cdot (3 (- 3))$

Cancel the common factor.

$y + 2 = - \frac{x}{3} + \frac{- 1}{3} \cdot (3 \cdot - 3)$

Rewrite the expression.

$y + 2 = - \frac{x}{3} - 1 \cdot - 3$

Multiply $- 1$ by $- 3$ .

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

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

Subtract $2$ from both sides of the equation.

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

Subtract $2$ from $3$ .

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

Step 5

Reorder terms.

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

Step 6

Remove parentheses.

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

Step 7

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

$y + 2 = - \frac{1}{3} \cdot (x - 9)$

Step 8

Find the Equation Using Point-Slope Formula (9,-2) and (-3,2)

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