Find the Equation Using Point-Slope Formula (-2,3) , (-10,9)

$(- 2, 3)$ , $(- 10, 9)$

Step 1

Find the slope of the line between $(- 2, 3)$ and $(- 10, 9)$ 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.

$m = \frac{9 - (3)}{- 10 - (- 2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{9 - 3}{- 10 - (- 2)}$

Subtract $3$ from $9$ .

$m = \frac{6}{- 10 - (- 2)}$

Simplify the denominator.

Multiply $- 1$ by $- 2$ .

$m = \frac{6}{- 10 + 2}$

Add $- 10$ and $2$ .

$m = \frac{6}{- 8}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $6$ and $- 8$ .

Factor $2$ out of $6$ .

$m = \frac{2 (3)}{- 8}$

Cancel the common factors.

Factor $2$ out of $- 8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{3}{4}$ and a given point $(- 2, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - \frac{3}{4} \cdot (x - (- 2))$

Step 3

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

$y - 3 = - \frac{3}{4} \cdot (x + 2)$

Step 4

Solve for $y$ .

Simplify $- \frac{3}{4} \cdot (x + 2)$ .

Rewrite.

$y - 3 = 0 + 0 - \frac{3}{4} \cdot (x + 2)$

Simplify terms.

Apply the distributive property.

$y - 3 = - \frac{3}{4} x - \frac{3}{4} \cdot 2$

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

$y - 3 = - \frac{x \cdot 3}{4} - \frac{3}{4} \cdot 2$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$y - 3 = - \frac{x \cdot 3}{4} + \frac{- 3}{4} \cdot 2$

Factor $2$ out of $4$ .

$y - 3 = - \frac{x \cdot 3}{4} + \frac{- 3}{2 (2)} \cdot 2$

Cancel the common factor.

$y - 3 = - \frac{x \cdot 3}{4} + \frac{- 3}{2 \cdot 2} \cdot 2$

Rewrite the expression.

$y - 3 = - \frac{x \cdot 3}{4} + \frac{- 3}{2}$

Simplify each term.

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

$y - 3 = - \frac{3 \cdot x}{4} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$y - 3 = - \frac{3 x}{4} - \frac{3}{2}$

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

Add $3$ to both sides of the equation.

$y = - \frac{3 x}{4} - \frac{3}{2} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{3 x}{4} - \frac{3}{2} + 3 \cdot \frac{2}{2}$

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

$y = - \frac{3 x}{4} - \frac{3}{2} + \frac{3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{4} + \frac{- 3 + 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $3$ by $2$ .

$y = - \frac{3 x}{4} + \frac{- 3 + 6}{2}$

Add $- 3$ and $6$ .

$y = - \frac{3 x}{4} + \frac{3}{2}$

Step 5

Reorder terms.

$y = - (\frac{3}{4} x) + \frac{3}{2}$

Step 6

Remove parentheses.

$y = - \frac{3}{4} x + \frac{3}{2}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{3}{4} x + \frac{3}{2}$

Point-slope form:

$y - 3 = - \frac{3}{4} \cdot (x + 2)$

Step 8

Find the Equation Using Point-Slope Formula (-2,3) , (-10,9)

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