Find the Equation Using Point-Slope Formula (2,5) , (9,2)

$(2, 5)$ , $(9, 2)$

Step 1

Find the slope of the line between $(2, 5)$ and $(9, 2)$ 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.

$m = \frac{2 - (5)}{9 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{2 - 5}{9 - (2)}$

Subtract $5$ from $2$ .

$m = \frac{- 3}{9 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{- 3}{9 - 2}$

Subtract $2$ from $9$ .

$m = \frac{- 3}{7}$

Move the negative in front of the fraction.

$m = - \frac{3}{7}$

Step 2

Use the slope $- \frac{3}{7}$ and a given point $(2, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = - \frac{3}{7} \cdot (x - (2))$

Step 3

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

$y - 5 = - \frac{3}{7} \cdot (x - 2)$

Step 4

Solve for $y$ .

Simplify $- \frac{3}{7} \cdot (x - 2)$ .

Rewrite.

$y - 5 = 0 + 0 - \frac{3}{7} \cdot (x - 2)$

Simplify by adding zeros.

$y - 5 = - \frac{3}{7} \cdot (x - 2)$

Apply the distributive property.

$y - 5 = - \frac{3}{7} x - \frac{3}{7} \cdot - 2$

Combine $x$ and $\frac{3}{7}$ .

$y - 5 = - \frac{x \cdot 3}{7} - \frac{3}{7} \cdot - 2$

Multiply $- \frac{3}{7} \cdot - 2$ .

Multiply $- 2$ by $- 1$ .

$y - 5 = - \frac{x \cdot 3}{7} + 2 (\frac{3}{7})$

Combine $2$ and $\frac{3}{7}$ .

$y - 5 = - \frac{x \cdot 3}{7} + \frac{2 \cdot 3}{7}$

Multiply $2$ by $3$ .

$y - 5 = - \frac{x \cdot 3}{7} + \frac{6}{7}$

Move $3$ to the left of $x$ .

$y - 5 = - \frac{3 x}{7} + \frac{6}{7}$

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

Add $5$ to both sides of the equation.

$y = - \frac{3 x}{7} + \frac{6}{7} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{3 x}{7} + \frac{6}{7} + 5 \cdot \frac{7}{7}$

Combine $5$ and $\frac{7}{7}$ .

$y = - \frac{3 x}{7} + \frac{6}{7} + \frac{5 \cdot 7}{7}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{7} + \frac{6 + 5 \cdot 7}{7}$

Simplify the numerator.

Multiply $5$ by $7$ .

$y = - \frac{3 x}{7} + \frac{6 + 35}{7}$

Add $6$ and $35$ .

$y = - \frac{3 x}{7} + \frac{41}{7}$

Step 5

Reorder terms.

$y = - (\frac{3}{7} x) + \frac{41}{7}$

Step 6

Remove parentheses.

$y = - \frac{3}{7} x + \frac{41}{7}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{3}{7} x + \frac{41}{7}$

Point-slope form:

$y - 5 = - \frac{3}{7} \cdot (x - 2)$

Step 8

Find the Equation Using Point-Slope Formula (2,5) , (9,2)

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