Find the Equation Using Point-Slope Formula (-2,6) , (2,14)

$(- 2, 6)$ , $(2, 14)$

Step 1

Find the slope of the line between $(- 2, 6)$ and $(2, 14)$ 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.

$m = \frac{14 - (6)}{2 - (- 2)}$

Simplify.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $14 - (6)$ and $2 - (- 2)$ .

Rewrite $14$ as $- 1 (- 14)$ .

$m = \frac{- 1 \cdot - 14 - (6)}{2 - (- 2)}$

Factor $- 1$ out of $- 1 (- 14) - (6)$ .

$m = \frac{- 1 (- 14 + 6)}{2 - (- 2)}$

Reorder terms.

$m = \frac{- 1 (- 14 + 6)}{2 - 2 \cdot - 1}$

Factor $2$ out of $- 1 (- 14 + 6)$ .

$m = \frac{2 (- 1 (- 7 + 3))}{2 - 2 \cdot - 1}$

Cancel the common factors.

Factor $2$ out of $2$ .

$m = \frac{2 (- 1 (- 7 + 3))}{2 (1) - 2 \cdot - 1}$

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

$m = \frac{2 (- 1 (- 7 + 3))}{2 (1) + 2 (1)}$

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

$m = \frac{2 (- 1 (- 7 + 3))}{2 (1 + 1)}$

Cancel the common factor.

$m = \frac{2 (- 1 (- 7 + 3))}{2 (1 + 1)}$

Rewrite the expression.

$m = \frac{- 1 (- 7 + 3)}{1 + 1}$

Add $- 7$ and $3$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

Simplify the expression.

Multiply $- 1$ by $- 4$ .

$m = \frac{4}{2}$

Divide $4$ by $2$ .

$m = 2$

Step 2

Use the slope $2$ and a given point $(- 2, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = 2 \cdot (x - (- 2))$

Step 3

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

$y - 6 = 2 \cdot (x + 2)$

Step 4

Solve for $y$ .

Simplify $2 \cdot (x + 2)$ .

Rewrite.

$y - 6 = 0 + 0 + 2 \cdot (x + 2)$

Simplify by adding zeros.

$y - 6 = 2 \cdot (x + 2)$

Apply the distributive property.

$y - 6 = 2 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$y - 6 = 2 x + 4$

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

Add $6$ to both sides of the equation.

$y = 2 x + 4 + 6$

Add $4$ and $6$ .

$y = 2 x + 10$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 2 x + 10$

Point-slope form:

$y - 6 = 2 \cdot (x + 2)$

Step 6

Find the Equation Using Point-Slope Formula (-2,6) , (2,14)

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