Find the Equation Using Point-Slope Formula (-7,21) , (5,-15)

$(- 7, 21)$ , $(5, - 15)$

Step 1

Find the slope of the line between $(- 7, 21)$ and $(5, - 15)$ 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.

$m = \frac{- 15 - (21)}{5 - (- 7)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $21$ .

$m = \frac{- 15 - 21}{5 - (- 7)}$

Subtract $21$ from $- 15$ .

$m = \frac{- 36}{5 - (- 7)}$

Simplify the denominator.

Multiply $- 1$ by $- 7$ .

$m = \frac{- 36}{5 + 7}$

Add $5$ and $7$ .

$m = \frac{- 36}{12}$

Divide $- 36$ by $12$ .

$m = - 3$

Step 2

Use the slope $- 3$ and a given point $(- 7, 21)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (21) = - 3 \cdot (x - (- 7))$

Step 3

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

$y - 21 = - 3 \cdot (x + 7)$

Step 4

Solve for $y$ .

Simplify $- 3 \cdot (x + 7)$ .

Rewrite.

$y - 21 = 0 + 0 - 3 \cdot (x + 7)$

Simplify by adding zeros.

$y - 21 = - 3 \cdot (x + 7)$

Apply the distributive property.

$y - 21 = - 3 x - 3 \cdot 7$

Multiply $- 3$ by $7$ .

$y - 21 = - 3 x - 21$

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

Add $21$ to both sides of the equation.

$y = - 3 x - 21 + 21$

Combine the opposite terms in $- 3 x - 21 + 21$ .

Add $- 21$ and $21$ .

$y = - 3 x + 0$

Add $- 3 x$ and $0$ .

$y = - 3 x$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 3 x$

Point-slope form:

$y - 21 = - 3 \cdot (x + 7)$

Step 6

Find the Equation Using Point-Slope Formula (-7,21) , (5,-15)

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