Find the Equation Using Point-Slope Formula (8,5) , (-12,-9)

$(8, 5)$ , $(- 12, - 9)$

Step 1

Find the slope of the line between $(8, 5)$ and $(- 12, - 9)$ 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.

$m = \frac{- 9 - (5)}{- 12 - (8)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{- 9 - 5}{- 12 - (8)}$

Subtract $5$ from $- 9$ .

$m = \frac{- 14}{- 12 - (8)}$

Simplify the denominator.

Multiply $- 1$ by $8$ .

$m = \frac{- 14}{- 12 - 8}$

Subtract $8$ from $- 12$ .

$m = \frac{- 14}{- 20}$

Cancel the common factor of $- 14$ and $- 20$ .

Factor $- 2$ out of $- 14$ .

$m = \frac{- 2 \cdot 7}{- 20}$

Cancel the common factors.

Factor $- 2$ out of $- 20$ .

$m = \frac{- 2 \cdot 7}{- 2 \cdot 10}$

Cancel the common factor.

$m = \frac{- 2 \cdot 7}{- 2 \cdot 10}$

Rewrite the expression.

$m = \frac{7}{10}$

Step 2

Use the slope $\frac{7}{10}$ and a given point $(8, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = \frac{7}{10} \cdot (x - (8))$

Step 3

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

$y - 5 = \frac{7}{10} \cdot (x - 8)$

Step 4

Solve for $y$ .

Simplify $\frac{7}{10} \cdot (x - 8)$ .

Rewrite.

$y - 5 = 0 + 0 + \frac{7}{10} \cdot (x - 8)$

Simplify by adding zeros.

$y - 5 = \frac{7}{10} \cdot (x - 8)$

Apply the distributive property.

$y - 5 = \frac{7}{10} x + \frac{7}{10} \cdot - 8$

Combine $\frac{7}{10}$ and $x$ .

$y - 5 = \frac{7 x}{10} + \frac{7}{10} \cdot - 8$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$y - 5 = \frac{7 x}{10} + \frac{7}{2 (5)} \cdot - 8$

Factor $2$ out of $- 8$ .

$y - 5 = \frac{7 x}{10} + \frac{7}{2 \cdot 5} \cdot (2 \cdot - 4)$

Cancel the common factor.

$y - 5 = \frac{7 x}{10} + \frac{7}{2 \cdot 5} \cdot (2 \cdot - 4)$

Rewrite the expression.

$y - 5 = \frac{7 x}{10} + \frac{7}{5} \cdot - 4$

Combine $\frac{7}{5}$ and $- 4$ .

$y - 5 = \frac{7 x}{10} + \frac{7 \cdot - 4}{5}$

Simplify the expression.

Multiply $7$ by $- 4$ .

$y - 5 = \frac{7 x}{10} + \frac{- 28}{5}$

Move the negative in front of the fraction.

$y - 5 = \frac{7 x}{10} - \frac{28}{5}$

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

Add $5$ to both sides of the equation.

$y = \frac{7 x}{10} - \frac{28}{5} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{7 x}{10} - \frac{28}{5} + 5 \cdot \frac{5}{5}$

Combine $5$ and $\frac{5}{5}$ .

$y = \frac{7 x}{10} - \frac{28}{5} + \frac{5 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = \frac{7 x}{10} + \frac{- 28 + 5 \cdot 5}{5}$

Simplify the numerator.

Multiply $5$ by $5$ .

$y = \frac{7 x}{10} + \frac{- 28 + 25}{5}$

Add $- 28$ and $25$ .

$y = \frac{7 x}{10} + \frac{- 3}{5}$

Move the negative in front of the fraction.

$y = \frac{7 x}{10} - \frac{3}{5}$

Step 5

Reorder terms.

$y = \frac{7}{10} x - \frac{3}{5}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{7}{10} x - \frac{3}{5}$

Point-slope form:

$y - 5 = \frac{7}{10} \cdot (x - 8)$

Step 7

Find the Equation Using Point-Slope Formula (8,5) , (-12,-9)

Related Questions

Support LCMGCF.com by disabling your adblocker.