Find the Equation Using Point-Slope Formula (6,8) , (3,-9)

$(6, 8)$ , $(3, - 9)$

Step 1

Find the slope of the line between $(6, 8)$ and $(3, - 9)$ 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.

$m = \frac{- 9 - (8)}{3 - (6)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $8$ .

$m = \frac{- 9 - 8}{3 - (6)}$

Subtract $8$ from $- 9$ .

$m = \frac{- 17}{3 - (6)}$

Simplify the denominator.

Multiply $- 1$ by $6$ .

$m = \frac{- 17}{3 - 6}$

Subtract $6$ from $3$ .

$m = \frac{- 17}{- 3}$

Dividing two negative values results in a positive value.

$m = \frac{17}{3}$

Step 2

Use the slope $\frac{17}{3}$ and a given point $(6, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = \frac{17}{3} \cdot (x - (6))$

Step 3

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

$y - 8 = \frac{17}{3} \cdot (x - 6)$

Step 4

Solve for $y$ .

Simplify $\frac{17}{3} \cdot (x - 6)$ .

Rewrite.

$y - 8 = 0 + 0 + \frac{17}{3} \cdot (x - 6)$

Simplify by adding zeros.

$y - 8 = \frac{17}{3} \cdot (x - 6)$

Apply the distributive property.

$y - 8 = \frac{17}{3} x + \frac{17}{3} \cdot - 6$

Combine $\frac{17}{3}$ and $x$ .

$y - 8 = \frac{17 x}{3} + \frac{17}{3} \cdot - 6$

Cancel the common factor of $3$ .

Factor $3$ out of $- 6$ .

$y - 8 = \frac{17 x}{3} + \frac{17}{3} \cdot (3 (- 2))$

Cancel the common factor.

$y - 8 = \frac{17 x}{3} + \frac{17}{3} \cdot (3 \cdot - 2)$

Rewrite the expression.

$y - 8 = \frac{17 x}{3} + 17 \cdot - 2$

Multiply $17$ by $- 2$ .

$y - 8 = \frac{17 x}{3} - 34$

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

Add $8$ to both sides of the equation.

$y = \frac{17 x}{3} - 34 + 8$

Add $- 34$ and $8$ .

$y = \frac{17 x}{3} - 26$

Step 5

Reorder terms.

$y = \frac{17}{3} x - 26$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{17}{3} x - 26$

Point-slope form:

$y - 8 = \frac{17}{3} \cdot (x - 6)$

Step 7

Find the Equation Using Point-Slope Formula (6,8) , (3,-9)

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