Find the Equation Using Point-Slope Formula (-2,-2) , (-3,5)

$(- 2, - 2)$ , $(- 3, 5)$

Step 1

Find the slope of the line between $(- 2, - 2)$ and $(- 3, 5)$ 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.

$m = \frac{5 - (- 2)}{- 3 - (- 2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{5 + 2}{- 3 - (- 2)}$

Add $5$ and $2$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 2$ .

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

Add $- 3$ and $2$ .

$m = \frac{7}{- 1}$

Divide $7$ by $- 1$ .

$m = - 7$

Step 2

Use the slope $- 7$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = - 7 \cdot (x - (- 2))$

Step 3

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

$y + 2 = - 7 \cdot (x + 2)$

Step 4

Solve for $y$ .

Simplify $- 7 \cdot (x + 2)$ .

Rewrite.

$y + 2 = 0 + 0 - 7 \cdot (x + 2)$

Simplify by adding zeros.

$y + 2 = - 7 \cdot (x + 2)$

Apply the distributive property.

$y + 2 = - 7 x - 7 \cdot 2$

Multiply $- 7$ by $2$ .

$y + 2 = - 7 x - 14$

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

Subtract $2$ from both sides of the equation.

$y = - 7 x - 14 - 2$

Subtract $2$ from $- 14$ .

$y = - 7 x - 16$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 7 x - 16$

Point-slope form:

$y + 2 = - 7 \cdot (x + 2)$

Step 6

Find the Equation Using Point-Slope Formula (-2,-2) , (-3,5)

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