Find the Equation Using Point-Slope Formula (-3,-23) , (5,9)

$(- 3, - 23)$ , $(5, 9)$

Step 1

Find the slope of the line between $(- 3, - 23)$ and $(5, 9)$ 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.

$m = \frac{9 - (- 23)}{5 - (- 3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 23$ .

$m = \frac{9 + 23}{5 - (- 3)}$

Add $9$ and $23$ .

$m = \frac{32}{5 - (- 3)}$

Simplify the denominator.

Multiply $- 1$ by $- 3$ .

$m = \frac{32}{5 + 3}$

Add $5$ and $3$ .

$m = \frac{32}{8}$

Divide $32$ by $8$ .

$m = 4$

Step 2

Use the slope $4$ and a given point $(- 3, - 23)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 23) = 4 \cdot (x - (- 3))$

Step 3

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

$y + 23 = 4 \cdot (x + 3)$

Step 4

Solve for $y$ .

Simplify $4 \cdot (x + 3)$ .

Rewrite.

$y + 23 = 0 + 0 + 4 \cdot (x + 3)$

Simplify by adding zeros.

$y + 23 = 4 \cdot (x + 3)$

Apply the distributive property.

$y + 23 = 4 x + 4 \cdot 3$

Multiply $4$ by $3$ .

$y + 23 = 4 x + 12$

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

Subtract $23$ from both sides of the equation.

$y = 4 x + 12 - 23$

Subtract $23$ from $12$ .

$y = 4 x - 11$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 4 x - 11$

Point-slope form:

$y + 23 = 4 \cdot (x + 3)$

Step 6

Find the Equation Using Point-Slope Formula (-3,-23) , (5,9)

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