Find the Equation Using Point-Slope Formula (12,6) , (4,4)

$(12, 6)$ , $(4, 4)$

Step 1

Find the slope of the line between $(12, 6)$ and $(4, 4)$ 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.

$m = \frac{4 - (6)}{4 - (12)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $6$ .

$m = \frac{4 - 6}{4 - (12)}$

Subtract $6$ from $4$ .

$m = \frac{- 2}{4 - (12)}$

Simplify the denominator.

Multiply $- 1$ by $12$ .

$m = \frac{- 2}{4 - 12}$

Subtract $12$ from $4$ .

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

Cancel the common factor of $- 2$ and $- 8$ .

Factor $- 2$ out of $- 2$ .

$m = \frac{- 2 \cdot 1}{- 8}$

Cancel the common factors.

Factor $- 2$ out of $- 8$ .

$m = \frac{- 2 \cdot 1}{- 2 \cdot 4}$

Cancel the common factor.

$m = \frac{- 2 \cdot 1}{- 2 \cdot 4}$

Rewrite the expression.

$m = \frac{1}{4}$

Step 2

Use the slope $\frac{1}{4}$ and a given point $(12, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = \frac{1}{4} \cdot (x - (12))$

Step 3

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

$y - 6 = \frac{1}{4} \cdot (x - 12)$

Step 4

Solve for $y$ .

Simplify $\frac{1}{4} \cdot (x - 12)$ .

Rewrite.

$y - 6 = 0 + 0 + \frac{1}{4} \cdot (x - 12)$

Simplify by adding zeros.

$y - 6 = \frac{1}{4} \cdot (x - 12)$

Apply the distributive property.

$y - 6 = \frac{1}{4} x + \frac{1}{4} \cdot - 12$

Combine $\frac{1}{4}$ and $x$ .

$y - 6 = \frac{x}{4} + \frac{1}{4} \cdot - 12$

Cancel the common factor of $4$ .

Factor $4$ out of $- 12$ .

$y - 6 = \frac{x}{4} + \frac{1}{4} \cdot (4 (- 3))$

Cancel the common factor.

$y - 6 = \frac{x}{4} + \frac{1}{4} \cdot (4 \cdot - 3)$

Rewrite the expression.

$y - 6 = \frac{x}{4} - 3$

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

Add $6$ to both sides of the equation.

$y = \frac{x}{4} - 3 + 6$

Add $- 3$ and $6$ .

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

Step 5

Reorder terms.

$y = \frac{1}{4} x + 3$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{1}{4} x + 3$

Point-slope form:

$y - 6 = \frac{1}{4} \cdot (x - 12)$

Step 7

Find the Equation Using Point-Slope Formula (12,6) , (4,4)

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