Find the Equation Using Point-Slope Formula (-5,14) , (1,-16)

$(- 5, 14)$ , $(1, - 16)$

Step 1

Find the slope of the line between $(- 5, 14)$ and $(1, - 16)$ 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.

$m = \frac{- 16 - (14)}{1 - (- 5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $14$ .

$m = \frac{- 16 - 14}{1 - (- 5)}$

Subtract $14$ from $- 16$ .

$m = \frac{- 30}{1 - (- 5)}$

Simplify the denominator.

Multiply $- 1$ by $- 5$ .

$m = \frac{- 30}{1 + 5}$

Add $1$ and $5$ .

$m = \frac{- 30}{6}$

Divide $- 30$ by $6$ .

$m = - 5$

Step 2

Use the slope $- 5$ and a given point $(- 5, 14)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (14) = - 5 \cdot (x - (- 5))$

Step 3

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

$y - 14 = - 5 \cdot (x + 5)$

Step 4

Solve for $y$ .

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

Rewrite.

$y - 14 = 0 + 0 - 5 \cdot (x + 5)$

Simplify by adding zeros.

$y - 14 = - 5 \cdot (x + 5)$

Apply the distributive property.

$y - 14 = - 5 x - 5 \cdot 5$

Multiply $- 5$ by $5$ .

$y - 14 = - 5 x - 25$

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

Add $14$ to both sides of the equation.

$y = - 5 x - 25 + 14$

Add $- 25$ and $14$ .

$y = - 5 x - 11$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 5 x - 11$

Point-slope form:

$y - 14 = - 5 \cdot (x + 5)$

Step 6

Find the Equation Using Point-Slope Formula (-5,14) , (1,-16)

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