Find the Equation Using Point-Slope Formula (-4,2) , (2,6)

$(- 4, 2)$ , $(2, 6)$

Step 1

Find the slope of the line between $(- 4, 2)$ and $(2, 6)$ 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.

$m = \frac{6 - (2)}{2 - (- 4)}$

Simplify.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $6 - (2)$ and $2 - (- 4)$ .

Rewrite $6$ as $- 1 (- 6)$ .

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

Factor $- 1$ out of $- 1 (- 6) - (2)$ .

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

Reorder terms.

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Factor $2$ out of $- 4 \cdot - 1$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Add $- 3$ and $1$ .

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

Simplify the denominator.

Multiply $- 2$ by $- 1$ .

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

Add $1$ and $2$ .

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

Multiply $- 1$ by $- 2$ .

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

Step 2

Use the slope $\frac{2}{3}$ and a given point $(- 4, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3

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

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

Step 4

Solve for $y$ .

Simplify $\frac{2}{3} \cdot (x + 4)$ .

Rewrite.

$y - 2 = 0 + 0 + \frac{2}{3} \cdot (x + 4)$

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Multiply $\frac{2}{3} \cdot 4$ .

Combine $\frac{2}{3}$ and $4$ .

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

Multiply $2$ by $4$ .

$y - 2 = \frac{2 x}{3} + \frac{8}{3}$

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

Add $2$ to both sides of the equation.

$y = \frac{2 x}{3} + \frac{8}{3} + 2$

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

$y = \frac{2 x}{3} + \frac{8}{3} + 2 \cdot \frac{3}{3}$

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

$y = \frac{2 x}{3} + \frac{8}{3} + \frac{2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{2 x}{3} + \frac{8 + 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$y = \frac{2 x}{3} + \frac{8 + 6}{3}$

Add $8$ and $6$ .

$y = \frac{2 x}{3} + \frac{14}{3}$

Step 5

Reorder terms.

$y = \frac{2}{3} x + \frac{14}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{2}{3} x + \frac{14}{3}$

Point-slope form:

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

Step 7

Find the Equation Using Point-Slope Formula (-4,2) , (2,6)

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