Find the Equation Using Point-Slope Formula (-4,10) , (-7,-5)

$(- 4, 10)$ , $(- 7, - 5)$

Step 1

Find the slope of the line between $(- 4, 10)$ and $(- 7, - 5)$ 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.

$m = \frac{- 5 - (10)}{- 7 - (- 4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $10$ .

$m = \frac{- 5 - 10}{- 7 - (- 4)}$

Subtract $10$ from $- 5$ .

$m = \frac{- 15}{- 7 - (- 4)}$

Simplify the denominator.

Multiply $- 1$ by $- 4$ .

$m = \frac{- 15}{- 7 + 4}$

Add $- 7$ and $4$ .

$m = \frac{- 15}{- 3}$

Divide $- 15$ by $- 3$ .

$m = 5$

Step 2

Use the slope $5$ and a given point $(- 4, 10)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (10) = 5 \cdot (x - (- 4))$

Step 3

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

$y - 10 = 5 \cdot (x + 4)$

Step 4

Solve for $y$ .

Simplify $5 \cdot (x + 4)$ .

Rewrite.

$y - 10 = 0 + 0 + 5 \cdot (x + 4)$

Simplify by adding zeros.

$y - 10 = 5 \cdot (x + 4)$

Apply the distributive property.

$y - 10 = 5 x + 5 \cdot 4$

Multiply $5$ by $4$ .

$y - 10 = 5 x + 20$

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

Add $10$ to both sides of the equation.

$y = 5 x + 20 + 10$

Add $20$ and $10$ .

$y = 5 x + 30$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 5 x + 30$

Point-slope form:

$y - 10 = 5 \cdot (x + 4)$

Step 6

Find the Equation Using Point-Slope Formula (-4,10) , (-7,-5)

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