Find the Equation Using Point-Slope Formula (4,1) , (12,5)

$(4, 1)$ , $(12, 5)$

Step 1

Find the slope of the line between $(4, 1)$ and $(12, 5)$ 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.

$m = \frac{5 - (1)}{12 - (4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $1$ .

$m = \frac{5 - 1}{12 - (4)}$

Subtract $1$ from $5$ .

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

Simplify the denominator.

Multiply $- 1$ by $4$ .

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

Subtract $4$ from $12$ .

$m = \frac{4}{8}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$m = \frac{4 (1)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$m = \frac{1}{2}$

Step 2

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

$y - (1) = \frac{1}{2} \cdot (x - (4))$

Step 3

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

$y - 1 = \frac{1}{2} \cdot (x - 4)$

Step 4

Solve for $y$ .

Simplify $\frac{1}{2} \cdot (x - 4)$ .

Rewrite.

$y - 1 = 0 + 0 + \frac{1}{2} \cdot (x - 4)$

Simplify by adding zeros.

$y - 1 = \frac{1}{2} \cdot (x - 4)$

Apply the distributive property.

$y - 1 = \frac{1}{2} x + \frac{1}{2} \cdot - 4$

Combine $\frac{1}{2}$ and $x$ .

$y - 1 = \frac{x}{2} + \frac{1}{2} \cdot - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y - 1 = \frac{x}{2} - 2$

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

Add $1$ to both sides of the equation.

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

Add $- 2$ and $1$ .

$y = \frac{x}{2} - 1$

Step 5

Reorder terms.

$y = \frac{1}{2} x - 1$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{1}{2} x - 1$

Point-slope form:

$y - 1 = \frac{1}{2} \cdot (x - 4)$

Step 7

Find the Equation Using Point-Slope Formula (4,1) , (12,5)

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