Find the Equation Using Point-Slope Formula (2,8) , (3,4)

$(2, 8)$ , $(3, 4)$

Step 1

Find the slope of the line between $(2, 8)$ and $(3, 4)$ 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.

$m = \frac{4 - (8)}{3 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $8$ .

$m = \frac{4 - 8}{3 - (2)}$

Subtract $8$ from $4$ .

$m = \frac{- 4}{3 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $3$ .

$m = \frac{- 4}{1}$

Divide $- 4$ by $1$ .

$m = - 4$

Step 2

Use the slope $- 4$ and a given point $(2, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = - 4 \cdot (x - (2))$

Step 3

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

$y - 8 = - 4 \cdot (x - 2)$

Step 4

Solve for $y$ .

Simplify $- 4 \cdot (x - 2)$ .

Rewrite.

$y - 8 = 0 + 0 - 4 \cdot (x - 2)$

Simplify by adding zeros.

$y - 8 = - 4 \cdot (x - 2)$

Apply the distributive property.

$y - 8 = - 4 x - 4 \cdot - 2$

Multiply $- 4$ by $- 2$ .

$y - 8 = - 4 x + 8$

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

Add $8$ to both sides of the equation.

$y = - 4 x + 8 + 8$

Add $8$ and $8$ .

$y = - 4 x + 16$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 4 x + 16$

Point-slope form:

$y - 8 = - 4 \cdot (x - 2)$

Step 6

Find the Equation Using Point-Slope Formula (2,8) , (3,4)

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