Find the Equation Using Point-Slope Formula (3,-2) and (5,-9)

$(3, - 2)$ and $(5, - 9)$

Step 1

Find the slope of the line between $(3, - 2)$ and $(5, - 9)$ 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.

$m = \frac{- 9 - (- 2)}{5 - (3)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{- 9 + 2}{5 - (3)}$

Add $- 9$ and $2$ .

$m = \frac{- 7}{5 - (3)}$

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{- 7}{5 - 3}$

Subtract $3$ from $5$ .

$m = \frac{- 7}{2}$

Move the negative in front of the fraction.

$m = - \frac{7}{2}$

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

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

Multiply $- 3$ by $- 1$ .

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

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

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

Multiply $3$ by $7$ .

$y + 2 = - \frac{x \cdot 7}{2} + \frac{21}{2}$

Move $7$ to the left of $x$ .

$y + 2 = - \frac{7 x}{2} + \frac{21}{2}$

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

Subtract $2$ from both sides of the equation.

$y = - \frac{7 x}{2} + \frac{21}{2} - 2$

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

$y = - \frac{7 x}{2} + \frac{21}{2} - 2 \cdot \frac{2}{2}$

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

$y = - \frac{7 x}{2} + \frac{21}{2} + \frac{- 2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{7 x}{2} + \frac{21 - 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$y = - \frac{7 x}{2} + \frac{21 - 4}{2}$

Subtract $4$ from $21$ .

$y = - \frac{7 x}{2} + \frac{17}{2}$

Step 5

Reorder terms.

$y = - (\frac{7}{2} x) + \frac{17}{2}$

Step 6

Remove parentheses.

$y = - \frac{7}{2} x + \frac{17}{2}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{7}{2} x + \frac{17}{2}$

Point-slope form:

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

Step 8

Find the Equation Using Point-Slope Formula (3,-2) and (5,-9)

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