Find the Equation Using Point-Slope Formula (2,-1) , (8,4)

$(2, - 1)$ , $(8, 4)$

Step 1

Find the slope of the line between $(2, - 1)$ and $(8, 4)$ 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.

$m = \frac{4 - (- 1)}{8 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 1$ .

$m = \frac{4 + 1}{8 - (2)}$

Add $4$ and $1$ .

$m = \frac{5}{8 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{5}{8 - 2}$

Subtract $2$ from $8$ .

$m = \frac{5}{6}$

Step 2

Use the slope $\frac{5}{6}$ and a given point $(2, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = \frac{5}{6} \cdot (x - (2))$

Step 3

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

$y + 1 = \frac{5}{6} \cdot (x - 2)$

Step 4

Solve for $y$ .

Simplify $\frac{5}{6} \cdot (x - 2)$ .

Rewrite.

$y + 1 = 0 + 0 + \frac{5}{6} \cdot (x - 2)$

Simplify by adding zeros.

$y + 1 = \frac{5}{6} \cdot (x - 2)$

Apply the distributive property.

$y + 1 = \frac{5}{6} x + \frac{5}{6} \cdot - 2$

Combine $\frac{5}{6}$ and $x$ .

$y + 1 = \frac{5 x}{6} + \frac{5}{6} \cdot - 2$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$y + 1 = \frac{5 x}{6} + \frac{5}{2 (3)} \cdot - 2$

Factor $2$ out of $- 2$ .

$y + 1 = \frac{5 x}{6} + \frac{5}{2 \cdot 3} \cdot (2 \cdot - 1)$

Cancel the common factor.

$y + 1 = \frac{5 x}{6} + \frac{5}{2 \cdot 3} \cdot (2 \cdot - 1)$

Rewrite the expression.

$y + 1 = \frac{5 x}{6} + \frac{5}{3} \cdot - 1$

Combine $\frac{5}{3}$ and $- 1$ .

$y + 1 = \frac{5 x}{6} + \frac{5 \cdot - 1}{3}$

Simplify the expression.

Multiply $5$ by $- 1$ .

$y + 1 = \frac{5 x}{6} + \frac{- 5}{3}$

Move the negative in front of the fraction.

$y + 1 = \frac{5 x}{6} - \frac{5}{3}$

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

Subtract $1$ from both sides of the equation.

$y = \frac{5 x}{6} - \frac{5}{3} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{5 x}{6} - \frac{5}{3} - 1 \cdot \frac{3}{3}$

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

$y = \frac{5 x}{6} - \frac{5}{3} + \frac{- 1 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{5 x}{6} + \frac{- 5 - 1 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 1$ by $3$ .

$y = \frac{5 x}{6} + \frac{- 5 - 3}{3}$

Subtract $3$ from $- 5$ .

$y = \frac{5 x}{6} + \frac{- 8}{3}$

Move the negative in front of the fraction.

$y = \frac{5 x}{6} - \frac{8}{3}$

Step 5

Reorder terms.

$y = \frac{5}{6} x - \frac{8}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{5}{6} x - \frac{8}{3}$

Point-slope form:

$y + 1 = \frac{5}{6} \cdot (x - 2)$

Step 7

Find the Equation Using Point-Slope Formula (2,-1) , (8,4)

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