Find the Equation Using Point-Slope Formula (-6,6) , (9,1)

$(- 6, 6)$ , $(9, 1)$

Step 1

Find the slope of the line between $(- 6, 6)$ and $(9, 1)$ 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.

$m = \frac{1 - (6)}{9 - (- 6)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $6$ .

$m = \frac{1 - 6}{9 - (- 6)}$

Subtract $6$ from $1$ .

$m = \frac{- 5}{9 - (- 6)}$

Simplify the denominator.

Multiply $- 1$ by $- 6$ .

$m = \frac{- 5}{9 + 6}$

Add $9$ and $6$ .

$m = \frac{- 5}{15}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 5$ and $15$ .

Factor $5$ out of $- 5$ .

$m = \frac{5 (- 1)}{15}$

Cancel the common factors.

Factor $5$ out of $15$ .

$m = \frac{5 \cdot - 1}{5 \cdot 3}$

Cancel the common factor.

$m = \frac{5 \cdot - 1}{5 \cdot 3}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 2

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

$y - (6) = - \frac{1}{3} \cdot (x - (- 6))$

Step 3

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

$y - 6 = - \frac{1}{3} \cdot (x + 6)$

Step 4

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x + 6)$ .

Rewrite.

$y - 6 = 0 + 0 - \frac{1}{3} \cdot (x + 6)$

Simplify by adding zeros.

$y - 6 = - \frac{1}{3} \cdot (x + 6)$

Apply the distributive property.

$y - 6 = - \frac{1}{3} x - \frac{1}{3} \cdot 6$

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

$y - 6 = - \frac{x}{3} - \frac{1}{3} \cdot 6$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y - 6 = - \frac{x}{3} + \frac{- 1}{3} \cdot 6$

Factor $3$ out of $6$ .

$y - 6 = - \frac{x}{3} + \frac{- 1}{3} \cdot (3 (2))$

Cancel the common factor.

$y - 6 = - \frac{x}{3} + \frac{- 1}{3} \cdot (3 \cdot 2)$

Rewrite the expression.

$y - 6 = - \frac{x}{3} - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$y - 6 = - \frac{x}{3} - 2$

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

Add $6$ to both sides of the equation.

$y = - \frac{x}{3} - 2 + 6$

Add $- 2$ and $6$ .

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

Step 5

Reorder terms.

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

Step 6

Remove parentheses.

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

Step 7

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

$y - 6 = - \frac{1}{3} \cdot (x + 6)$

Step 8

Find the Equation Using Point-Slope Formula (-6,6) , (9,1)

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