Find the Equation Using Point-Slope Formula (1,3) , (5,1)

$(1, 3)$ , $(5, 1)$

Step 1

Find the slope of the line between $(1, 3)$ and $(5, 1)$ 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.

$m = \frac{1 - (3)}{5 - (1)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{1 - 3}{5 - (1)}$

Subtract $3$ from $1$ .

$m = \frac{- 2}{5 - (1)}$

Simplify the denominator.

Multiply $- 1$ by $1$ .

$m = \frac{- 2}{5 - 1}$

Subtract $1$ from $5$ .

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2$ .

$m = \frac{2 (- 1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 2

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

$y - (3) = - \frac{1}{2} \cdot (x - (1))$

Step 3

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

$y - 3 = - \frac{1}{2} \cdot (x - 1)$

Step 4

Solve for $y$ .

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

Rewrite.

$y - 3 = 0 + 0 - \frac{1}{2} \cdot (x - 1)$

Simplify by adding zeros.

$y - 3 = - \frac{1}{2} \cdot (x - 1)$

Apply the distributive property.

$y - 3 = - \frac{1}{2} x - \frac{1}{2} \cdot - 1$

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

$y - 3 = - \frac{x}{2} - \frac{1}{2} \cdot - 1$

Multiply $- \frac{1}{2} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$y - 3 = - \frac{x}{2} + 1 (\frac{1}{2})$

Multiply $\frac{1}{2}$ by $1$ .

$y - 3 = - \frac{x}{2} + \frac{1}{2}$

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

Add $3$ to both sides of the equation.

$y = - \frac{x}{2} + \frac{1}{2} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x}{2} + \frac{1}{2} + 3 \cdot \frac{2}{2}$

Combine $3$ and $\frac{2}{2}$ .

$y = - \frac{x}{2} + \frac{1}{2} + \frac{3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{x}{2} + \frac{1 + 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $3$ by $2$ .

$y = - \frac{x}{2} + \frac{1 + 6}{2}$

Add $1$ and $6$ .

$y = - \frac{x}{2} + \frac{7}{2}$

Step 5

Reorder terms.

$y = - (\frac{1}{2} x) + \frac{7}{2}$

Step 6

Remove parentheses.

$y = - \frac{1}{2} x + \frac{7}{2}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{1}{2} x + \frac{7}{2}$

Point-slope form:

$y - 3 = - \frac{1}{2} \cdot (x - 1)$

Step 8

Find the Equation Using Point-Slope Formula (1,3) , (5,1)

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