Find the Equation Using Point-Slope Formula (-2,-3) , (4,5)

$(- 2, - 3)$ , $(4, 5)$

Step 1

Find the slope of the line between $(- 2, - 3)$ and $(4, 5)$ 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.

$m = \frac{5 - (- 3)}{4 - (- 2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 3$ .

$m = \frac{5 + 3}{4 - (- 2)}$

Add $5$ and $3$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 2$ .

$m = \frac{8}{4 + 2}$

Add $4$ and $2$ .

$m = \frac{8}{6}$

Cancel the common factor of $8$ and $6$ .

Factor $2$ out of $8$ .

$m = \frac{2 (4)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

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

$y - (- 3) = \frac{4}{3} \cdot (x - (- 2))$

Step 3

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

$y + 3 = \frac{4}{3} \cdot (x + 2)$

Step 4

Solve for $y$ .

Simplify $\frac{4}{3} \cdot (x + 2)$ .

Rewrite.

$y + 3 = 0 + 0 + \frac{4}{3} \cdot (x + 2)$

Simplify by adding zeros.

$y + 3 = \frac{4}{3} \cdot (x + 2)$

Apply the distributive property.

$y + 3 = \frac{4}{3} x + \frac{4}{3} \cdot 2$

Combine $\frac{4}{3}$ and $x$ .

$y + 3 = \frac{4 x}{3} + \frac{4}{3} \cdot 2$

Multiply $\frac{4}{3} \cdot 2$ .

Combine $\frac{4}{3}$ and $2$ .

$y + 3 = \frac{4 x}{3} + \frac{4 \cdot 2}{3}$

Multiply $4$ by $2$ .

$y + 3 = \frac{4 x}{3} + \frac{8}{3}$

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

Subtract $3$ from both sides of the equation.

$y = \frac{4 x}{3} + \frac{8}{3} - 3$

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

$y = \frac{4 x}{3} + \frac{8}{3} - 3 \cdot \frac{3}{3}$

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

$y = \frac{4 x}{3} + \frac{8}{3} + \frac{- 3 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{4 x}{3} + \frac{8 - 3 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 3$ by $3$ .

$y = \frac{4 x}{3} + \frac{8 - 9}{3}$

Subtract $9$ from $8$ .

$y = \frac{4 x}{3} + \frac{- 1}{3}$

Move the negative in front of the fraction.

$y = \frac{4 x}{3} - \frac{1}{3}$

Step 5

Reorder terms.

$y = \frac{4}{3} x - \frac{1}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{4}{3} x - \frac{1}{3}$

Point-slope form:

$y + 3 = \frac{4}{3} \cdot (x + 2)$

Step 7

Find the Equation Using Point-Slope Formula (-2,-3) , (4,5)

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