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

$(- 5, - 5)$ , $(5, 9)$

Step 1

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

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 5$ .

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

Add $9$ and $5$ .

$m = \frac{14}{5 - (- 5)}$

Simplify the denominator.

Multiply $- 1$ by $- 5$ .

$m = \frac{14}{5 + 5}$

Add $5$ and $5$ .

$m = \frac{14}{10}$

Cancel the common factor of $14$ and $10$ .

Factor $2$ out of $14$ .

$m = \frac{2 (7)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$m = \frac{2 \cdot 7}{2 \cdot 5}$

Cancel the common factor.

$m = \frac{2 \cdot 7}{2 \cdot 5}$

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

$y + 5 = \frac{7 x}{5} + 7$

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

Subtract $5$ from both sides of the equation.

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

Subtract $5$ from $7$ .

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

Step 5

Reorder terms.

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

Step 6

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 7

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

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