Find the Equation Using Point-Slope Formula (-4,2) , (3,-4)

$(- 4, 2)$ , $(3, - 4)$

Step 1

Find the slope of the line between $(- 4, 2)$ and $(3, - 4)$ 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.

$m = \frac{- 4 - (2)}{3 - (- 4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $2$ .

$m = \frac{- 4 - 2}{3 - (- 4)}$

Subtract $2$ from $- 4$ .

$m = \frac{- 6}{3 - (- 4)}$

Simplify the denominator.

Multiply $- 1$ by $- 4$ .

$m = \frac{- 6}{3 + 4}$

Add $3$ and $4$ .

$m = \frac{- 6}{7}$

Move the negative in front of the fraction.

$m = - \frac{6}{7}$

Step 2

Use the slope $- \frac{6}{7}$ and a given point $(- 4, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{6}{7} \cdot (x - (- 4))$

Step 3

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

$y - 2 = - \frac{6}{7} \cdot (x + 4)$

Step 4

Solve for $y$ .

Simplify $- \frac{6}{7} \cdot (x + 4)$ .

Rewrite.

$y - 2 = 0 + 0 - \frac{6}{7} \cdot (x + 4)$

Simplify by adding zeros.

$y - 2 = - \frac{6}{7} \cdot (x + 4)$

Apply the distributive property.

$y - 2 = - \frac{6}{7} x - \frac{6}{7} \cdot 4$

Combine $x$ and $\frac{6}{7}$ .

$y - 2 = - \frac{x \cdot 6}{7} - \frac{6}{7} \cdot 4$

Multiply $- \frac{6}{7} \cdot 4$ .

Multiply $4$ by $- 1$ .

$y - 2 = - \frac{x \cdot 6}{7} - 4 (\frac{6}{7})$

Combine $- 4$ and $\frac{6}{7}$ .

$y - 2 = - \frac{x \cdot 6}{7} + \frac{- 4 \cdot 6}{7}$

Multiply $- 4$ by $6$ .

$y - 2 = - \frac{x \cdot 6}{7} + \frac{- 24}{7}$

Simplify each term.

Move $6$ to the left of $x$ .

$y - 2 = - \frac{6 \cdot x}{7} + \frac{- 24}{7}$

Move the negative in front of the fraction.

$y - 2 = - \frac{6 x}{7} - \frac{24}{7}$

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

Add $2$ to both sides of the equation.

$y = - \frac{6 x}{7} - \frac{24}{7} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{6 x}{7} - \frac{24}{7} + 2 \cdot \frac{7}{7}$

Combine $2$ and $\frac{7}{7}$ .

$y = - \frac{6 x}{7} - \frac{24}{7} + \frac{2 \cdot 7}{7}$

Combine the numerators over the common denominator.

$y = - \frac{6 x}{7} + \frac{- 24 + 2 \cdot 7}{7}$

Simplify the numerator.

Multiply $2$ by $7$ .

$y = - \frac{6 x}{7} + \frac{- 24 + 14}{7}$

Add $- 24$ and $14$ .

$y = - \frac{6 x}{7} + \frac{- 10}{7}$

Move the negative in front of the fraction.

$y = - \frac{6 x}{7} - \frac{10}{7}$

Step 5

Reorder terms.

$y = - (\frac{6}{7} x) - \frac{10}{7}$

Step 6

Remove parentheses.

$y = - \frac{6}{7} x - \frac{10}{7}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{6}{7} x - \frac{10}{7}$

Point-slope form:

$y - 2 = - \frac{6}{7} \cdot (x + 4)$

Step 8

Find the Equation Using Point-Slope Formula (-4,2) , (3,-4)

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