Find the Equation Using Point-Slope Formula (3,-8) , (6,-4)

$(3, - 8)$ , $(6, - 4)$

Step 1

Find the slope of the line between $(3, - 8)$ and $(6, - 4)$ 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.

$m = \frac{- 4 - (- 8)}{6 - (3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 8$ .

$m = \frac{- 4 + 8}{6 - (3)}$

Add $- 4$ and $8$ .

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

Simplify the denominator.

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $6$ .

$m = \frac{4}{3}$

Step 2

Use the slope $\frac{4}{3}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 8) = \frac{4}{3} \cdot (x - (3))$

Step 3

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

$y + 8 = \frac{4}{3} \cdot (x - 3)$

Step 4

Solve for $y$ .

Simplify $\frac{4}{3} \cdot (x - 3)$ .

Rewrite.

$y + 8 = 0 + 0 + \frac{4}{3} \cdot (x - 3)$

Simplify by adding zeros.

$y + 8 = \frac{4}{3} \cdot (x - 3)$

Apply the distributive property.

$y + 8 = \frac{4}{3} x + \frac{4}{3} \cdot - 3$

Combine $\frac{4}{3}$ and $x$ .

$y + 8 = \frac{4 x}{3} + \frac{4}{3} \cdot - 3$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

$y + 8 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 (- 1))$

Cancel the common factor.

$y + 8 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 \cdot - 1)$

Rewrite the expression.

$y + 8 = \frac{4 x}{3} + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$y + 8 = \frac{4 x}{3} - 4$

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

Subtract $8$ from both sides of the equation.

$y = \frac{4 x}{3} - 4 - 8$

Subtract $8$ from $- 4$ .

$y = \frac{4 x}{3} - 12$

Step 5

Reorder terms.

$y = \frac{4}{3} x - 12$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{4}{3} x - 12$

Point-slope form:

$y + 8 = \frac{4}{3} \cdot (x - 3)$

Step 7

Find the Equation Using Point-Slope Formula (3,-8) , (6,-4)

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