Find the Equation Using Point-Slope Formula (-3,2) , (5,8)

$(- 3, 2)$ , $(5, 8)$

Step 1

Find the slope of the line between $(- 3, 2)$ and $(5, 8)$ 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.

$m = \frac{8 - (2)}{5 - (- 3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $2$ .

$m = \frac{8 - 2}{5 - (- 3)}$

Subtract $2$ from $8$ .

$m = \frac{6}{5 - (- 3)}$

Simplify the denominator.

Multiply $- 1$ by $- 3$ .

$m = \frac{6}{5 + 3}$

Add $5$ and $3$ .

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

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

Factor $2$ out of $6$ .

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

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

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

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

Multiply $3$ by $3$ .

$y - 2 = \frac{3 x}{4} + \frac{9}{4}$

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

Add $2$ to both sides of the equation.

$y = \frac{3 x}{4} + \frac{9}{4} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{3 x}{4} + \frac{9}{4} + 2 \cdot \frac{4}{4}$

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

$y = \frac{3 x}{4} + \frac{9}{4} + \frac{2 \cdot 4}{4}$

Combine the numerators over the common denominator.

$y = \frac{3 x}{4} + \frac{9 + 2 \cdot 4}{4}$

Simplify the numerator.

Multiply $2$ by $4$ .

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

Add $9$ and $8$ .

$y = \frac{3 x}{4} + \frac{17}{4}$

Step 5

Reorder terms.

$y = \frac{3}{4} x + \frac{17}{4}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{3}{4} x + \frac{17}{4}$

Point-slope form:

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

Step 7

Find the Equation Using Point-Slope Formula (-3,2) , (5,8)

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