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

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

Step 1

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

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

Simplify.

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

Rewrite $0$ as $- 1 (0)$ .

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

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

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

Reorder terms.

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

Factor $2$ out of $4$ .

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

Factor $2$ out of $- 4 \cdot - 1$ .

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Multiply $- 2$ by $- 1$ .

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

Add $2$ and $2$ .

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

Simplify the expression.

Add $0$ and $1$ .

$m = \frac{4}{- 1 \cdot 1}$

Multiply $- 1$ by $1$ .

$m = \frac{4}{- 1}$

Divide $4$ by $- 1$ .

$m = - 4$

Step 2

Use the slope $- 4$ and a given point $(2, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = - 4 \cdot (x - (2))$

Step 3

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

$y + 4 = - 4 \cdot (x - 2)$

Step 4

Solve for $y$ .

Simplify $- 4 \cdot (x - 2)$ .

Rewrite.

$y + 4 = 0 + 0 - 4 \cdot (x - 2)$

Simplify by adding zeros.

$y + 4 = - 4 \cdot (x - 2)$

Apply the distributive property.

$y + 4 = - 4 x - 4 \cdot - 2$

Multiply $- 4$ by $- 2$ .

$y + 4 = - 4 x + 8$

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

Subtract $4$ from both sides of the equation.

$y = - 4 x + 8 - 4$

Subtract $4$ from $8$ .

$y = - 4 x + 4$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 4 x + 4$

Point-slope form:

$y + 4 = - 4 \cdot (x - 2)$

Step 6

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

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