Find the Equation Using Point-Slope Formula (-3,6) , (2,-7)

$(- 3, 6)$ , $(2, - 7)$

Step 1

Find the slope of the line between $(- 3, 6)$ and $(2, - 7)$ 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.

$m = \frac{- 7 - (6)}{2 - (- 3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $6$ .

$m = \frac{- 7 - 6}{2 - (- 3)}$

Subtract $6$ from $- 7$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 3$ .

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

Add $2$ and $3$ .

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

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{13}{5}$ and a given point $(- 3, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = - \frac{13}{5} \cdot (x - (- 3))$

Step 3

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

$y - 6 = - \frac{13}{5} \cdot (x + 3)$

Step 4

Solve for $y$ .

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

Rewrite.

$y - 6 = 0 + 0 - \frac{13}{5} \cdot (x + 3)$

Simplify by adding zeros.

$y - 6 = - \frac{13}{5} \cdot (x + 3)$

Apply the distributive property.

$y - 6 = - \frac{13}{5} x - \frac{13}{5} \cdot 3$

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

$y - 6 = - \frac{x \cdot 13}{5} - \frac{13}{5} \cdot 3$

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

Multiply $3$ by $- 1$ .

$y - 6 = - \frac{x \cdot 13}{5} - 3 (\frac{13}{5})$

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

$y - 6 = - \frac{x \cdot 13}{5} + \frac{- 3 \cdot 13}{5}$

Multiply $- 3$ by $13$ .

$y - 6 = - \frac{x \cdot 13}{5} + \frac{- 39}{5}$

Simplify each term.

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

$y - 6 = - \frac{13 \cdot x}{5} + \frac{- 39}{5}$

Move the negative in front of the fraction.

$y - 6 = - \frac{13 x}{5} - \frac{39}{5}$

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

Add $6$ to both sides of the equation.

$y = - \frac{13 x}{5} - \frac{39}{5} + 6$

To write $6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{13 x}{5} - \frac{39}{5} + 6 \cdot \frac{5}{5}$

Combine $6$ and $\frac{5}{5}$ .

$y = - \frac{13 x}{5} - \frac{39}{5} + \frac{6 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = - \frac{13 x}{5} + \frac{- 39 + 6 \cdot 5}{5}$

Simplify the numerator.

Multiply $6$ by $5$ .

$y = - \frac{13 x}{5} + \frac{- 39 + 30}{5}$

Add $- 39$ and $30$ .

$y = - \frac{13 x}{5} + \frac{- 9}{5}$

Move the negative in front of the fraction.

$y = - \frac{13 x}{5} - \frac{9}{5}$

Step 5

Reorder terms.

$y = - (\frac{13}{5} x) - \frac{9}{5}$

Step 6

Remove parentheses.

$y = - \frac{13}{5} x - \frac{9}{5}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{13}{5} x - \frac{9}{5}$

Point-slope form:

$y - 6 = - \frac{13}{5} \cdot (x + 3)$

Step 8

Find the Equation Using Point-Slope Formula (-3,6) , (2,-7)

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