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

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

Step 1

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

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

Simplify.

Reduce the expression by cancelling the common factors.

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

Rewrite $4$ as $- 1 (- 4)$ .

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

Factor $- 1$ out of $- 1 (- 4) - (- 2)$ .

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

Reorder terms.

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

Factor $2$ out of $- 1 (- 4 - 2)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Factor $2$ out of $- 6 \cdot - 1$ .

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

Factor $2$ out of $2 (1) + 2 (- 3 \cdot - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Subtract $1$ from $- 2$ .

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

Simplify the denominator.

Multiply $- 3$ by $- 1$ .

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

Add $1$ and $3$ .

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

Multiply $- 1$ by $- 3$ .

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

Step 2

Use the slope $\frac{3}{4}$ and a given point $(- 6, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = \frac{3}{4} \cdot (x - (- 6))$

Step 3

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

$y + 2 = \frac{3}{4} \cdot (x + 6)$

Step 4

Solve for $y$ .

Simplify $\frac{3}{4} \cdot (x + 6)$ .

Rewrite.

$y + 2 = 0 + 0 + \frac{3}{4} \cdot (x + 6)$

Simplify by adding zeros.

$y + 2 = \frac{3}{4} \cdot (x + 6)$

Apply the distributive property.

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$y + 2 = \frac{3 x}{4} + \frac{3}{2 (2)} \cdot 6$

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Multiply $3$ by $3$ .

$y + 2 = \frac{3 x}{4} + \frac{9}{2}$

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

Subtract $2$ from both sides of the equation.

$y = \frac{3 x}{4} + \frac{9}{2} - 2$

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

$y = \frac{3 x}{4} + \frac{9}{2} - 2 \cdot \frac{2}{2}$

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

$y = \frac{3 x}{4} + \frac{9}{2} + \frac{- 2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{3 x}{4} + \frac{9 - 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$y = \frac{3 x}{4} + \frac{9 - 4}{2}$

Subtract $4$ from $9$ .

$y = \frac{3 x}{4} + \frac{5}{2}$

Step 5

Reorder terms.

$y = \frac{3}{4} x + \frac{5}{2}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{3}{4} x + \frac{5}{2}$

Point-slope form:

$y + 2 = \frac{3}{4} \cdot (x + 6)$

Step 7

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

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