Find the Equation Using Point-Slope Formula (3,12) , (1,4)

$(3, 12)$ , $(1, 4)$

Step 1

Find the slope of the line between $(3, 12)$ and $(1, 4)$ 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.

$m = \frac{4 - (12)}{1 - (3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $12$ .

$m = \frac{4 - 12}{1 - (3)}$

Subtract $12$ from $4$ .

$m = \frac{- 8}{1 - (3)}$

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{- 8}{1 - 3}$

Subtract $3$ from $1$ .

$m = \frac{- 8}{- 2}$

Divide $- 8$ by $- 2$ .

$m = 4$

Step 2

Use the slope $4$ and a given point $(3, 12)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (12) = 4 \cdot (x - (3))$

Step 3

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

$y - 12 = 4 \cdot (x - 3)$

Step 4

Solve for $y$ .

Simplify $4 \cdot (x - 3)$ .

Rewrite.

$y - 12 = 0 + 0 + 4 \cdot (x - 3)$

Simplify by adding zeros.

$y - 12 = 4 \cdot (x - 3)$

Apply the distributive property.

$y - 12 = 4 x + 4 \cdot - 3$

Multiply $4$ by $- 3$ .

$y - 12 = 4 x - 12$

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

Add $12$ to both sides of the equation.

$y = 4 x - 12 + 12$

Combine the opposite terms in $4 x - 12 + 12$ .

Add $- 12$ and $12$ .

$y = 4 x + 0$

Add $4 x$ and $0$ .

$y = 4 x$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 4 x$

Point-slope form:

$y - 12 = 4 \cdot (x - 3)$

Step 6

Find the Equation Using Point-Slope Formula (3,12) , (1,4)

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