Find the Equation Using Point-Slope Formula (-5,3) , (7,-9)

$(- 5, 3)$ , $(7, - 9)$

Step 1

Find the slope of the line between $(- 5, 3)$ and $(7, - 9)$ 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.

$m = \frac{- 9 - (3)}{7 - (- 5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{- 9 - 3}{7 - (- 5)}$

Subtract $3$ from $- 9$ .

$m = \frac{- 12}{7 - (- 5)}$

Simplify the denominator.

Multiply $- 1$ by $- 5$ .

$m = \frac{- 12}{7 + 5}$

Add $7$ and $5$ .

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

Divide $- 12$ by $12$ .

$m = - 1$

Step 2

Use the slope $- 1$ and a given point $(- 5, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - 1 \cdot (x - (- 5))$

Step 3

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

$y - 3 = - 1 \cdot (x + 5)$

Step 4

Solve for $y$ .

Simplify $- 1 \cdot (x + 5)$ .

Rewrite.

$y - 3 = 0 + 0 - 1 \cdot (x + 5)$

Simplify by adding zeros.

$y - 3 = - 1 \cdot (x + 5)$

Apply the distributive property.

$y - 3 = - 1 x - 1 \cdot 5$

Simplify the expression.

Rewrite $- 1 x$ as $- x$ .

$y - 3 = - x - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$y - 3 = - x - 5$

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

Add $3$ to both sides of the equation.

$y = - x - 5 + 3$

Add $- 5$ and $3$ .

$y = - x - 2$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - x - 2$

Point-slope form:

$y - 3 = - 1 \cdot (x + 5)$

Step 6

Find the Equation Using Point-Slope Formula (-5,3) , (7,-9)

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