Find the Equation Using Point-Slope Formula (5,-2) , (-16,4)

$(5, - 2)$ , $(- 16, 4)$

Step 1

Find the slope of the line between $(5, - 2)$ and $(- 16, 4)$ 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.

$m = \frac{4 - (- 2)}{- 16 - (5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{4 + 2}{- 16 - (5)}$

Add $4$ and $2$ .

$m = \frac{6}{- 16 - (5)}$

Simplify the denominator.

Multiply $- 1$ by $5$ .

$m = \frac{6}{- 16 - 5}$

Subtract $5$ from $- 16$ .

$m = \frac{6}{- 21}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $6$ and $- 21$ .

Factor $3$ out of $6$ .

$m = \frac{3 (2)}{- 21}$

Cancel the common factors.

Factor $3$ out of $- 21$ .

$m = \frac{3 \cdot 2}{3 \cdot - 7}$

Cancel the common factor.

$m = \frac{3 \cdot 2}{3 \cdot - 7}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{2}{7}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3

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

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

Step 4

Solve for $y$ .

Simplify $- \frac{2}{7} \cdot (x - 5)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Multiply $- \frac{2}{7} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

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

Combine $5$ and $\frac{2}{7}$ .

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

Multiply $5$ by $2$ .

$y + 2 = - \frac{x \cdot 2}{7} + \frac{10}{7}$

Move $2$ to the left of $x$ .

$y + 2 = - \frac{2 x}{7} + \frac{10}{7}$

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

Subtract $2$ from both sides of the equation.

$y = - \frac{2 x}{7} + \frac{10}{7} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{2 x}{7} + \frac{10}{7} - 2 \cdot \frac{7}{7}$

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

$y = - \frac{2 x}{7} + \frac{10}{7} + \frac{- 2 \cdot 7}{7}$

Combine the numerators over the common denominator.

$y = - \frac{2 x}{7} + \frac{10 - 2 \cdot 7}{7}$

Simplify the numerator.

Multiply $- 2$ by $7$ .

$y = - \frac{2 x}{7} + \frac{10 - 14}{7}$

Subtract $14$ from $10$ .

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

Move the negative in front of the fraction.

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

Step 5

Reorder terms.

$y = - (\frac{2}{7} x) - \frac{4}{7}$

Step 6

Remove parentheses.

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

Step 7

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 8

Find the Equation Using Point-Slope Formula (5,-2) , (-16,4)

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