Find the Equation Using Point-Slope Formula (-4,-2) , (4,10)

$(- 4, - 2)$ , $(4, 10)$

Step 1

Find the slope of the line between $(- 4, - 2)$ and $(4, 10)$ 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.

$m = \frac{10 - (- 2)}{4 - (- 4)}$

Simplify.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $10 - (- 2)$ and $4 - (- 4)$ .

Rewrite $10$ as $- 1 (- 10)$ .

$m = \frac{- 1 \cdot - 10 - (- 2)}{4 - (- 4)}$

Factor $- 1$ out of $- 1 (- 10) - (- 2)$ .

$m = \frac{- 1 (- 10 - 2)}{4 - (- 4)}$

Reorder terms.

$m = \frac{- 1 (- 10 - 2)}{4 - 4 \cdot - 1}$

Factor $2$ out of $- 1 (- 10 - 2)$ .

$m = \frac{2 (- 1 (- 5 - 1))}{4 - 4 \cdot - 1}$

Cancel the common factors.

Factor $2$ out of $4$ .

$m = \frac{2 (- 1 (- 5 - 1))}{2 (2) - 4 \cdot - 1}$

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

$m = \frac{2 (- 1 (- 5 - 1))}{2 (2) + 2 (- 2 \cdot - 1)}$

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

$m = \frac{2 (- 1 (- 5 - 1))}{2 (2 - 2 \cdot - 1)}$

Cancel the common factor.

$m = \frac{2 (- 1 (- 5 - 1))}{2 (2 - 2 \cdot - 1)}$

Rewrite the expression.

$m = \frac{- 1 (- 5 - 1)}{2 - 2 \cdot - 1}$

Subtract $1$ from $- 5$ .

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

Simplify the denominator.

Multiply $- 2$ by $- 1$ .

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

Add $2$ and $2$ .

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

Reduce the expression by cancelling the common factors.

Multiply $- 1$ by $- 6$ .

$m = \frac{6}{4}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$m = \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Use the slope $\frac{3}{2}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $2$ .

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

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

Subtract $2$ from both sides of the equation.

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

Subtract $2$ from $6$ .

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

Step 5

Reorder terms.

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

Step 6

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 7

Find the Equation Using Point-Slope Formula (-4,-2) , (4,10)

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