Find the Equation Using Point-Slope Formula (3,-8) , (-2,5)

$(3, - 8)$ , $(- 2, 5)$

Step 1

Find the slope of the line between $(3, - 8)$ and $(- 2, 5)$ 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.

$m = \frac{5 - (- 8)}{- 2 - (3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 8$ .

$m = \frac{5 + 8}{- 2 - (3)}$

Add $5$ and $8$ .

$m = \frac{13}{- 2 - (3)}$

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{13}{- 2 - 3}$

Subtract $3$ from $- 2$ .

$m = \frac{13}{- 5}$

Move the negative in front of the fraction.

$m = - \frac{13}{5}$

Step 2

Use the slope $- \frac{13}{5}$ and a given point $(3, - 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 8) = - \frac{13}{5} \cdot (x - (3))$

Step 3

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

$y + 8 = - \frac{13}{5} \cdot (x - 3)$

Step 4

Solve for $y$ .

Simplify $- \frac{13}{5} \cdot (x - 3)$ .

Rewrite.

$y + 8 = 0 + 0 - \frac{13}{5} \cdot (x - 3)$

Simplify by adding zeros.

$y + 8 = - \frac{13}{5} \cdot (x - 3)$

Apply the distributive property.

$y + 8 = - \frac{13}{5} x - \frac{13}{5} \cdot - 3$

Combine $x$ and $\frac{13}{5}$ .

$y + 8 = - \frac{x \cdot 13}{5} - \frac{13}{5} \cdot - 3$

Multiply $- \frac{13}{5} \cdot - 3$ .

Multiply $- 3$ by $- 1$ .

$y + 8 = - \frac{x \cdot 13}{5} + 3 (\frac{13}{5})$

Combine $3$ and $\frac{13}{5}$ .

$y + 8 = - \frac{x \cdot 13}{5} + \frac{3 \cdot 13}{5}$

Multiply $3$ by $13$ .

$y + 8 = - \frac{x \cdot 13}{5} + \frac{39}{5}$

Move $13$ to the left of $x$ .

$y + 8 = - \frac{13 x}{5} + \frac{39}{5}$

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

Subtract $8$ from both sides of the equation.

$y = - \frac{13 x}{5} + \frac{39}{5} - 8$

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{13 x}{5} + \frac{39}{5} - 8 \cdot \frac{5}{5}$

Combine $- 8$ and $\frac{5}{5}$ .

$y = - \frac{13 x}{5} + \frac{39}{5} + \frac{- 8 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = - \frac{13 x}{5} + \frac{39 - 8 \cdot 5}{5}$

Simplify the numerator.

Multiply $- 8$ by $5$ .

$y = - \frac{13 x}{5} + \frac{39 - 40}{5}$

Subtract $40$ from $39$ .

$y = - \frac{13 x}{5} + \frac{- 1}{5}$

Move the negative in front of the fraction.

$y = - \frac{13 x}{5} - \frac{1}{5}$

Step 5

Reorder terms.

$y = - (\frac{13}{5} x) - \frac{1}{5}$

Step 6

Remove parentheses.

$y = - \frac{13}{5} x - \frac{1}{5}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{13}{5} x - \frac{1}{5}$

Point-slope form:

$y + 8 = - \frac{13}{5} \cdot (x - 3)$

Step 8

Find the Equation Using Point-Slope Formula (3,-8) , (-2,5)

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