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

$(- 2, 2)$ , $(4, - 2)$

Step 1

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

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

Simplify.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 2 - (2)$ and $4 - (- 2)$ .

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

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

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

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

Reorder terms.

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

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

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Add $1$ and $1$ .

$m = \frac{- 1 \cdot 2}{2 + 1}$

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

$m = \frac{- 1 \cdot 2}{2 + 1}$

Add $2$ and $1$ .

$m = \frac{- 1 \cdot 2}{3}$

Simplify the expression.

Multiply $- 1$ by $2$ .

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

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

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

Multiply $2$ by $- 1$ .

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

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

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

Multiply $- 2$ by $2$ .

$y - 2 = - \frac{x \cdot 2}{3} + \frac{- 4}{3}$

Simplify each term.

Move $2$ to the left of $x$ .

$y - 2 = - \frac{2 \cdot x}{3} + \frac{- 4}{3}$

Move the negative in front of the fraction.

$y - 2 = - \frac{2 x}{3} - \frac{4}{3}$

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

Add $2$ to both sides of the equation.

$y = - \frac{2 x}{3} - \frac{4}{3} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{2 x}{3} - \frac{4}{3} + 2 \cdot \frac{3}{3}$

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

$y = - \frac{2 x}{3} - \frac{4}{3} + \frac{2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{2 x}{3} + \frac{- 4 + 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$y = - \frac{2 x}{3} + \frac{- 4 + 6}{3}$

Add $- 4$ and $6$ .

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

Step 5

Reorder terms.

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

Step 6

Remove parentheses.

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

Step 7

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 8

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

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