Find the Equation Using Point-Slope Formula (-10,-7) , (-5,-9)

$(- 10, - 7)$ , $(- 5, - 9)$

Step 1

Find the slope of the line between $(- 10, - 7)$ and $(- 5, - 9)$ 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.

$m = \frac{- 9 - (- 7)}{- 5 - (- 10)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 7$ .

$m = \frac{- 9 + 7}{- 5 - (- 10)}$

Add $- 9$ and $7$ .

$m = \frac{- 2}{- 5 - (- 10)}$

Simplify the denominator.

Multiply $- 1$ by $- 10$ .

$m = \frac{- 2}{- 5 + 10}$

Add $- 5$ and $10$ .

$m = \frac{- 2}{5}$

Move the negative in front of the fraction.

$m = - \frac{2}{5}$

Step 2

Use the slope $- \frac{2}{5}$ and a given point $(- 10, - 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 7) = - \frac{2}{5} \cdot (x - (- 10))$

Step 3

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

$y + 7 = - \frac{2}{5} \cdot (x + 10)$

Step 4

Solve for $y$ .

Simplify $- \frac{2}{5} \cdot (x + 10)$ .

Rewrite.

$y + 7 = 0 + 0 - \frac{2}{5} \cdot (x + 10)$

Simplify terms.

Apply the distributive property.

$y + 7 = - \frac{2}{5} x - \frac{2}{5} \cdot 10$

Combine $x$ and $\frac{2}{5}$ .

$y + 7 = - \frac{x \cdot 2}{5} - \frac{2}{5} \cdot 10$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$y + 7 = - \frac{x \cdot 2}{5} + \frac{- 2}{5} \cdot 10$

Factor $5$ out of $10$ .

$y + 7 = - \frac{x \cdot 2}{5} + \frac{- 2}{5} \cdot (5 (2))$

Cancel the common factor.

$y + 7 = - \frac{x \cdot 2}{5} + \frac{- 2}{5} \cdot (5 \cdot 2)$

Rewrite the expression.

$y + 7 = - \frac{x \cdot 2}{5} - 2 \cdot 2$

Multiply $- 2$ by $2$ .

$y + 7 = - \frac{x \cdot 2}{5} - 4$

Move $2$ to the left of $x$ .

$y + 7 = - \frac{2 x}{5} - 4$

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

Subtract $7$ from both sides of the equation.

$y = - \frac{2 x}{5} - 4 - 7$

Subtract $7$ from $- 4$ .

$y = - \frac{2 x}{5} - 11$

Step 5

Reorder terms.

$y = - (\frac{2}{5} x) - 11$

Step 6

Remove parentheses.

$y = - \frac{2}{5} x - 11$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{2}{5} x - 11$

Point-slope form:

$y + 7 = - \frac{2}{5} \cdot (x + 10)$

Step 8

Find the Equation Using Point-Slope Formula (-10,-7) , (-5,-9)

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