Find the Equation Using Point-Slope Formula (0,-2) , (4,-5)

$(0, - 2)$ , $(4, - 5)$

Step 1

Find the slope of the line between $(0, - 2)$ and $(4, - 5)$ 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.

$m = \frac{- 5 - (- 2)}{4 - (0)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{- 5 + 2}{4 - (0)}$

Add $- 5$ and $2$ .

$m = \frac{- 3}{4 - (0)}$

Simplify the denominator.

Multiply $- 1$ by $0$ .

$m = \frac{- 3}{4 + 0}$

Add $4$ and $0$ .

$m = \frac{- 3}{4}$

Move the negative in front of the fraction.

$m = - \frac{3}{4}$

Step 2

Use the slope $- \frac{3}{4}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = - \frac{3}{4} \cdot (x - (0))$

Step 3

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

$y + 2 = - \frac{3}{4} \cdot (x + 0)$

Step 4

Solve for $y$ .

Simplify $- \frac{3}{4} \cdot (x + 0)$ .

Add $x$ and $0$ .

$y + 2 = - \frac{3}{4} \cdot x$

Combine $x$ and $\frac{3}{4}$ .

$y + 2 = - \frac{x \cdot 3}{4}$

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

$y + 2 = - \frac{3 x}{4}$

Subtract $2$ from both sides of the equation.

$y = - \frac{3 x}{4} - 2$

Step 5

Reorder terms.

$y = - (\frac{3}{4} x) - 2$

Step 6

Remove parentheses.

$y = - \frac{3}{4} x - 2$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{3}{4} x - 2$

Point-slope form:

$y + 2 = - \frac{3}{4} \cdot (x + 0)$

Step 8

Find the Equation Using Point-Slope Formula (0,-2) , (4,-5)

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