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

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

Step 1

Find the slope of the line between $(- 2, 2)$ 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.

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $4$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 2$ .

$m = \frac{2}{3 + 2}$

Add $3$ and $2$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $\frac{2}{5}$ and $x$ .

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

Multiply $\frac{2}{5} \cdot 2$ .

Combine $\frac{2}{5}$ and $2$ .

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

Multiply $2$ by $2$ .

$y - 2 = \frac{2 x}{5} + \frac{4}{5}$

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

Add $2$ to both sides of the equation.

$y = \frac{2 x}{5} + \frac{4}{5} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{2 x}{5} + \frac{4}{5} + 2 \cdot \frac{5}{5}$

Combine $2$ and $\frac{5}{5}$ .

$y = \frac{2 x}{5} + \frac{4}{5} + \frac{2 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = \frac{2 x}{5} + \frac{4 + 2 \cdot 5}{5}$

Simplify the numerator.

Multiply $2$ by $5$ .

$y = \frac{2 x}{5} + \frac{4 + 10}{5}$

Add $4$ and $10$ .

$y = \frac{2 x}{5} + \frac{14}{5}$

Step 5

Reorder terms.

$y = \frac{2}{5} x + \frac{14}{5}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{2}{5} x + \frac{14}{5}$

Point-slope form:

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

Step 7

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

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