Find the Equation Using Point-Slope Formula (3,7) , (5,13)

$(3, 7)$ , $(5, 13)$

Step 1

Find the slope of the line between $(3, 7)$ and $(5, 13)$ 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.

$m = \frac{13 - (7)}{5 - (3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $7$ .

$m = \frac{13 - 7}{5 - (3)}$

Subtract $7$ from $13$ .

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

Simplify the denominator.

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $5$ .

$m = \frac{6}{2}$

Divide $6$ by $2$ .

$m = 3$

Step 2

Use the slope $3$ and a given point $(3, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = 3 \cdot (x - (3))$

Step 3

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

$y - 7 = 3 \cdot (x - 3)$

Step 4

Solve for $y$ .

Simplify $3 \cdot (x - 3)$ .

Rewrite.

$y - 7 = 0 + 0 + 3 \cdot (x - 3)$

Simplify by adding zeros.

$y - 7 = 3 \cdot (x - 3)$

Apply the distributive property.

$y - 7 = 3 x + 3 \cdot - 3$

Multiply $3$ by $- 3$ .

$y - 7 = 3 x - 9$

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

Add $7$ to both sides of the equation.

$y = 3 x - 9 + 7$

Add $- 9$ and $7$ .

$y = 3 x - 2$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 3 x - 2$

Point-slope form:

$y - 7 = 3 \cdot (x - 3)$

Step 6

Find the Equation Using Point-Slope Formula (3,7) , (5,13)

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