Find the Equation Using Point-Slope Formula (-3,2) , (6,3)

$(- 3, 2)$ , $(6, 3)$

Step 1

Find the slope of the line between $(- 3, 2)$ and $(6, 3)$ 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.

$m = \frac{3 - (2)}{6 - (- 3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $2$ .

$m = \frac{3 - 2}{6 - (- 3)}$

Subtract $2$ from $3$ .

$m = \frac{1}{6 - (- 3)}$

Simplify the denominator.

Multiply $- 1$ by $- 3$ .

$m = \frac{1}{6 + 3}$

Add $6$ and $3$ .

$m = \frac{1}{9}$

Step 2

Use the slope $\frac{1}{9}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = \frac{1}{9} \cdot (x - (- 3))$

Step 3

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

$y - 2 = \frac{1}{9} \cdot (x + 3)$

Step 4

Solve for $y$ .

Simplify $\frac{1}{9} \cdot (x + 3)$ .

Rewrite.

$y - 2 = 0 + 0 + \frac{1}{9} \cdot (x + 3)$

Simplify by adding zeros.

$y - 2 = \frac{1}{9} \cdot (x + 3)$

Apply the distributive property.

$y - 2 = \frac{1}{9} x + \frac{1}{9} \cdot 3$

Combine $\frac{1}{9}$ and $x$ .

$y - 2 = \frac{x}{9} + \frac{1}{9} \cdot 3$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$y - 2 = \frac{x}{9} + \frac{1}{3 (3)} \cdot 3$

Cancel the common factor.

$y - 2 = \frac{x}{9} + \frac{1}{3 \cdot 3} \cdot 3$

Rewrite the expression.

$y - 2 = \frac{x}{9} + \frac{1}{3}$

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

Add $2$ to both sides of the equation.

$y = \frac{x}{9} + \frac{1}{3} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x}{9} + \frac{1}{3} + 2 \cdot \frac{3}{3}$

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

$y = \frac{x}{9} + \frac{1}{3} + \frac{2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{x}{9} + \frac{1 + 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$y = \frac{x}{9} + \frac{1 + 6}{3}$

Add $1$ and $6$ .

$y = \frac{x}{9} + \frac{7}{3}$

Step 5

Reorder terms.

$y = \frac{1}{9} x + \frac{7}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{1}{9} x + \frac{7}{3}$

Point-slope form:

$y - 2 = \frac{1}{9} \cdot (x + 3)$

Step 7

Find the Equation Using Point-Slope Formula (-3,2) , (6,3)

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