Find the Equation Using Point-Slope Formula (5,2) , (-1,4)

$(5, 2)$ , $(- 1, 4)$

Step 1

Find the slope of the line between $(5, 2)$ 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.

$m = \frac{4 - (2)}{- 1 - (5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $2$ .

$m = \frac{4 - 2}{- 1 - (5)}$

Subtract $2$ from $4$ .

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

Simplify the denominator.

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $- 1$ .

$m = \frac{2}{- 6}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2$ and $- 6$ .

Factor $2$ out of $2$ .

$m = \frac{2 (1)}{- 6}$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$m = \frac{2 \cdot 1}{2 \cdot - 3}$

Cancel the common factor.

$m = \frac{2 \cdot 1}{2 \cdot - 3}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{1}{3}$ and a given point $(5, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{1}{3} \cdot (x - (5))$

Step 3

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

$y - 2 = - \frac{1}{3} \cdot (x - 5)$

Step 4

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x - 5)$ .

Rewrite.

$y - 2 = 0 + 0 - \frac{1}{3} \cdot (x - 5)$

Simplify by adding zeros.

$y - 2 = - \frac{1}{3} \cdot (x - 5)$

Apply the distributive property.

$y - 2 = - \frac{1}{3} x - \frac{1}{3} \cdot - 5$

Combine $x$ and $\frac{1}{3}$ .

$y - 2 = - \frac{x}{3} - \frac{1}{3} \cdot - 5$

Multiply $- \frac{1}{3} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$y - 2 = - \frac{x}{3} + 5 (\frac{1}{3})$

Combine $5$ and $\frac{1}{3}$ .

$y - 2 = - \frac{x}{3} + \frac{5}{3}$

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

Add $2$ to both sides of the equation.

$y = - \frac{x}{3} + \frac{5}{3} + 2$

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

$y = - \frac{x}{3} + \frac{5}{3} + 2 \cdot \frac{3}{3}$

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

$y = - \frac{x}{3} + \frac{5}{3} + \frac{2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{x}{3} + \frac{5 + 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$y = - \frac{x}{3} + \frac{5 + 6}{3}$

Add $5$ and $6$ .

$y = - \frac{x}{3} + \frac{11}{3}$

Step 5

Reorder terms.

$y = - (\frac{1}{3} x) + \frac{11}{3}$

Step 6

Remove parentheses.

$y = - \frac{1}{3} x + \frac{11}{3}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{1}{3} x + \frac{11}{3}$

Point-slope form:

$y - 2 = - \frac{1}{3} \cdot (x - 5)$

Step 8

Find the Equation Using Point-Slope Formula (5,2) , (-1,4)

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