Find the Equation Using Point-Slope Formula (3,-1) , (-3,-3)

$(3, - 1)$ , $(- 3, - 3)$

Step 1

Find the slope of the line between $(3, - 1)$ and $(- 3, - 3)$ 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.

$m = \frac{- 3 - (- 1)}{- 3 - (3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 1$ .

$m = \frac{- 3 + 1}{- 3 - (3)}$

Add $- 3$ and $1$ .

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

Simplify the denominator.

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $- 3$ .

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

Cancel the common factor of $- 2$ and $- 6$ .

Factor $- 2$ out of $- 2$ .

$m = \frac{- 2 \cdot 1}{- 6}$

Cancel the common factors.

Factor $- 2$ out of $- 6$ .

$m = \frac{- 2 \cdot 1}{- 2 \cdot 3}$

Cancel the common factor.

$m = \frac{- 2 \cdot 1}{- 2 \cdot 3}$

Rewrite the expression.

$m = \frac{1}{3}$

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

Simplify $\frac{1}{3} \cdot (x - 3)$ .

Rewrite.

$y + 1 = 0 + 0 + \frac{1}{3} \cdot (x - 3)$

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $\frac{1}{3}$ and $x$ .

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

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Subtract $1$ from both sides of the equation.

$y = \frac{x}{3} - 1 - 1$

Subtract $1$ from $- 1$ .

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

Step 5

Reorder terms.

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

Step 6

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 7

Find the Equation Using Point-Slope Formula (3,-1) , (-3,-3)

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