Find the Equation Using Point-Slope Formula (-6,7) , (2,-5)

$(- 6, 7)$ , $(2, - 5)$

Step 1

Find the slope of the line between $(- 6, 7)$ and $(2, - 5)$ 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.

$m = \frac{- 5 - (7)}{2 - (- 6)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $7$ .

$m = \frac{- 5 - 7}{2 - (- 6)}$

Subtract $7$ from $- 5$ .

$m = \frac{- 12}{2 - (- 6)}$

Simplify the denominator.

Multiply $- 1$ by $- 6$ .

$m = \frac{- 12}{2 + 6}$

Add $2$ and $6$ .

$m = \frac{- 12}{8}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 12$ and $8$ .

Factor $4$ out of $- 12$ .

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

Cancel the common factors.

Factor $4$ out of $8$ .

$m = \frac{4 \cdot - 3}{4 \cdot 2}$

Cancel the common factor.

$m = \frac{4 \cdot - 3}{4 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{3}{2}$ and a given point $(- 6, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = - \frac{3}{2} \cdot (x - (- 6))$

Step 3

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

$y - 7 = - \frac{3}{2} \cdot (x + 6)$

Step 4

Solve for $y$ .

Simplify $- \frac{3}{2} \cdot (x + 6)$ .

Rewrite.

$y - 7 = 0 + 0 - \frac{3}{2} \cdot (x + 6)$

Simplify terms.

Apply the distributive property.

$y - 7 = - \frac{3}{2} x - \frac{3}{2} \cdot 6$

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

$y - 7 = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot 6$

Cancel the common factor of $2$ .

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

$y - 7 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot 6$

Factor $2$ out of $6$ .

$y - 7 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 (3))$

Cancel the common factor.

$y - 7 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 \cdot 3)$

Rewrite the expression.

$y - 7 = - \frac{x \cdot 3}{2} - 3 \cdot 3$

Multiply $- 3$ by $3$ .

$y - 7 = - \frac{x \cdot 3}{2} - 9$

Move $3$ to the left of $x$ .

$y - 7 = - \frac{3 x}{2} - 9$

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

Add $7$ to both sides of the equation.

$y = - \frac{3 x}{2} - 9 + 7$

Add $- 9$ and $7$ .

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

Step 5

Reorder terms.

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

Step 6

Remove parentheses.

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

Step 7

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

$y - 7 = - \frac{3}{2} \cdot (x + 6)$

Step 8

Find the Equation Using Point-Slope Formula (-6,7) , (2,-5)

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