Find the Equation Using Point-Slope Formula (-3,7) , (2,-3)

$(- 3, 7)$ , $(2, - 3)$

Step 1

Find the slope of the line between $(- 3, 7)$ and $(2, - 3)$ 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.

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $7$ .

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

Subtract $7$ from $- 3$ .

$m = \frac{- 10}{2 - (- 3)}$

Simplify the denominator.

Multiply $- 1$ by $- 3$ .

$m = \frac{- 10}{2 + 3}$

Add $2$ and $3$ .

$m = \frac{- 10}{5}$

Divide $- 10$ by $5$ .

$m = - 2$

Step 2

Use the slope $- 2$ and a given point $(- 3, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = - 2 \cdot (x - (- 3))$

Step 3

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

$y - 7 = - 2 \cdot (x + 3)$

Step 4

Solve for $y$ .

Simplify $- 2 \cdot (x + 3)$ .

Rewrite.

$y - 7 = 0 + 0 - 2 \cdot (x + 3)$

Simplify by adding zeros.

$y - 7 = - 2 \cdot (x + 3)$

Apply the distributive property.

$y - 7 = - 2 x - 2 \cdot 3$

Multiply $- 2$ by $3$ .

$y - 7 = - 2 x - 6$

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

Add $7$ to both sides of the equation.

$y = - 2 x - 6 + 7$

Add $- 6$ and $7$ .

$y = - 2 x + 1$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 2 x + 1$

Point-slope form:

$y - 7 = - 2 \cdot (x + 3)$

Step 6

Find the Equation Using Point-Slope Formula (-3,7) , (2,-3)

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