Find the Equation Using Point-Slope Formula (-5,-2) , (4,-5)

$(- 5, - 2)$ , $(4, - 5)$

Step 1

Find the slope of the line between $(- 5, - 2)$ and $(4, - 5)$ 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.

$m = \frac{- 5 - (- 2)}{4 - (- 5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{- 5 + 2}{4 - (- 5)}$

Add $- 5$ and $2$ .

$m = \frac{- 3}{4 - (- 5)}$

Simplify the denominator.

Multiply $- 1$ by $- 5$ .

$m = \frac{- 3}{4 + 5}$

Add $4$ and $5$ .

$m = \frac{- 3}{9}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 3$ and $9$ .

Factor $3$ out of $- 3$ .

$m = \frac{3 (- 1)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$m = \frac{3 \cdot - 1}{3 \cdot 3}$

Cancel the common factor.

$m = \frac{3 \cdot - 1}{3 \cdot 3}$

Rewrite the expression.

$m = \frac{- 1}{3}$

Move the negative in front of the fraction.

$m = - \frac{1}{3}$

Step 2

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

$y - (- 2) = - \frac{1}{3} \cdot (x - (- 5))$

Step 3

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

$y + 2 = - \frac{1}{3} \cdot (x + 5)$

Step 4

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x + 5)$ .

Rewrite.

$y + 2 = 0 + 0 - \frac{1}{3} \cdot (x + 5)$

Simplify by adding zeros.

$y + 2 = - \frac{1}{3} \cdot (x + 5)$

Apply the distributive property.

$y + 2 = - \frac{1}{3} x - \frac{1}{3} \cdot 5$

Combine $x$ and $\frac{1}{3}$ .

$y + 2 = - \frac{x}{3} - \frac{1}{3} \cdot 5$

Multiply $- \frac{1}{3} \cdot 5$ .

Multiply $5$ by $- 1$ .

$y + 2 = - \frac{x}{3} - 5 (\frac{1}{3})$

Combine $- 5$ and $\frac{1}{3}$ .

$y + 2 = - \frac{x}{3} + \frac{- 5}{3}$

Move the negative in front of the fraction.

$y + 2 = - \frac{x}{3} - \frac{5}{3}$

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

Subtract $2$ from both sides of the equation.

$y = - \frac{x}{3} - \frac{5}{3} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{x}{3} - \frac{5}{3} - 2 \cdot \frac{3}{3}$

Combine $- 2$ and $\frac{3}{3}$ .

$y = - \frac{x}{3} - \frac{5}{3} + \frac{- 2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{x}{3} + \frac{- 5 - 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$y = - \frac{x}{3} + \frac{- 5 - 6}{3}$

Subtract $6$ from $- 5$ .

$y = - \frac{x}{3} + \frac{- 11}{3}$

Move the negative in front of the fraction.

$y = - \frac{x}{3} - \frac{11}{3}$

Step 5

Reorder terms.

$y = - (\frac{1}{3} x) - \frac{11}{3}$

Step 6

Remove parentheses.

$y = - \frac{1}{3} x - \frac{11}{3}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{1}{3} x - \frac{11}{3}$

Point-slope form:

$y + 2 = - \frac{1}{3} \cdot (x + 5)$

Step 8

Find the Equation Using Point-Slope Formula (-5,-2) , (4,-5)

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