Find the Equation Using Point-Slope Formula (4,2) , (7,8)

$(4, 2)$ , $(7, 8)$

Step 1

Find the slope of the line between $(4, 2)$ and $(7, 8)$ 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.

$m = \frac{8 - (2)}{7 - (4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $2$ .

$m = \frac{8 - 2}{7 - (4)}$

Subtract $2$ from $8$ .

$m = \frac{6}{7 - (4)}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$m = \frac{6}{7 - 4}$

Subtract $4$ from $7$ .

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

Divide $6$ by $3$ .

$m = 2$

Step 2

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

$y - (2) = 2 \cdot (x - (4))$

Step 3

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

$y - 2 = 2 \cdot (x - 4)$

Step 4

Solve for $y$ .

Simplify $2 \cdot (x - 4)$ .

Rewrite.

$y - 2 = 0 + 0 + 2 \cdot (x - 4)$

Simplify by adding zeros.

$y - 2 = 2 \cdot (x - 4)$

Apply the distributive property.

$y - 2 = 2 x + 2 \cdot - 4$

Multiply $2$ by $- 4$ .

$y - 2 = 2 x - 8$

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

Add $2$ to both sides of the equation.

$y = 2 x - 8 + 2$

Add $- 8$ and $2$ .

$y = 2 x - 6$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 2 x - 6$

Point-slope form:

$y - 2 = 2 \cdot (x - 4)$

Step 6

Find the Equation Using Point-Slope Formula (4,2) , (7,8)

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