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

$(5, 2)$ , $(2, 5)$

Step 1

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

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $5$ .

$m = \frac{3}{2 - (5)}$

Simplify the denominator.

Multiply $- 1$ by $5$ .

$m = \frac{3}{2 - 5}$

Subtract $5$ from $2$ .

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

Divide $3$ by $- 3$ .

$m = - 1$

Step 2

Use the slope $- 1$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - 1 \cdot (x - (5))$

Step 3

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

$y - 2 = - 1 \cdot (x - 5)$

Step 4

Solve for $y$ .

Simplify $- 1 \cdot (x - 5)$ .

Rewrite.

$y - 2 = 0 + 0 - 1 \cdot (x - 5)$

Simplify by adding zeros.

$y - 2 = - 1 \cdot (x - 5)$

Apply the distributive property.

$y - 2 = - 1 x - 1 \cdot - 5$

Simplify the expression.

Rewrite $- 1 x$ as $- x$ .

$y - 2 = - x - 1 \cdot - 5$

Multiply $- 1$ by $- 5$ .

$y - 2 = - x + 5$

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

Add $2$ to both sides of the equation.

$y = - x + 5 + 2$

Add $5$ and $2$ .

$y = - x + 7$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - x + 7$

Point-slope form:

$y - 2 = - 1 \cdot (x - 5)$

Step 6

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

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