Find the Equation Using Point-Slope Formula (3,-2) , (6,-4)

$(3, - 2)$ , $(6, - 4)$

Step 1

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

$m = \frac{- 4 - (- 2)}{6 - (3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{- 4 + 2}{6 - (3)}$

Add $- 4$ and $2$ .

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

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{- 2}{6 - 3}$

Subtract $3$ from $6$ .

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

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

Simplify $- \frac{2}{3} \cdot (x - 3)$ .

Rewrite.

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

Simplify terms.

Apply the distributive property.

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

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

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2}{3}$ into the numerator.

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 2$ by $- 1$ .

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

Move $2$ to the left of $x$ .

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

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

Subtract $2$ from both sides of the equation.

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

Combine the opposite terms in $- \frac{2 x}{3} + 2 - 2$ .

Subtract $2$ from $2$ .

$y = - \frac{2 x}{3} + 0$

Add $- \frac{2 x}{3}$ and $0$ .

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

Step 5

Reorder terms.

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

Step 6

Remove parentheses.

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

Step 7

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 8

Find the Equation Using Point-Slope Formula (3,-2) , (6,-4)

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