Find the Equation Using Point-Slope Formula (0,-2) , (-4,4)

$(0, - 2)$ , $(- 4, 4)$

Step 1

Find the slope of the line between $(0, - 2)$ and $(- 4, 4)$ 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.

$m = \frac{4 - (- 2)}{- 4 - (0)}$

Simplify.

Cancel the common factor of $4 - (- 2)$ and $- 4 - (0)$ .

Rewrite $- 4$ as $- 1 (4)$ .

$m = \frac{4 - (- 2)}{- 1 \cdot 4 - (0)}$

Factor $- 1$ out of $- 1 (4) - (0)$ .

$m = \frac{4 - (- 2)}{- 1 (4 + 0)}$

Reorder terms.

$m = \frac{4 - 2 \cdot - 1}{- 1 (4 + 0)}$

Factor $2$ out of $4$ .

$m = \frac{2 (2) - 2 \cdot - 1}{- 1 (4 + 0)}$

Factor $2$ out of $- 2 \cdot - 1$ .

$m = \frac{2 (2) + 2 (1)}{- 1 (4 + 0)}$

Factor $2$ out of $2 (2) + 2 (- - 1)$ .

$m = \frac{2 (2 + 1)}{- 1 (4 + 0)}$

Cancel the common factors.

Factor $2$ out of $- 1 (4 + 0)$ .

$m = \frac{2 (2 + 1)}{2 (- 1 (2 + 0))}$

Cancel the common factor.

$m = \frac{2 (2 + 1)}{2 (- 1 (2 + 0))}$

Rewrite the expression.

$m = \frac{2 + 1}{- 1 (2 + 0)}$

Simplify the numerator.

Multiply $- 1$ by $- 1$ .

$m = \frac{2 + 1}{- 1 (2 + 0)}$

Add $2$ and $1$ .

$m = \frac{3}{- 1 (2 + 0)}$

Simplify the expression.

Add $2$ and $0$ .

$m = \frac{3}{- 1 \cdot 2}$

Multiply $- 1$ by $2$ .

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

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{3}{2}$ and a given point $(0, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3

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

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

Step 4

Solve for $y$ .

Simplify $- \frac{3}{2} \cdot (x + 0)$ .

Add $x$ and $0$ .

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

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

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

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

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

Subtract $2$ from both sides of the equation.

$y = - \frac{3 x}{2} - 2$

Step 5

Reorder terms.

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

Step 6

Remove parentheses.

$y = - \frac{3}{2} x - 2$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{3}{2} x - 2$

Point-slope form:

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

Step 8

Find the Equation Using Point-Slope Formula (0,-2) , (-4,4)

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