Find the Equation Using Point-Slope Formula (4,0) , (-2,1)

$(4, 0)$ , $(- 2, 1)$

Step 1

Find the slope of the line between $(4, 0)$ and $(- 2, 1)$ 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.

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $0$ .

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

Add $1$ and $0$ .

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

Simplify the denominator.

Multiply $- 1$ by $4$ .

$m = \frac{1}{- 2 - 4}$

Subtract $4$ from $- 2$ .

$m = \frac{1}{- 6}$

Move the negative in front of the fraction.

$m = - \frac{1}{6}$

Step 2

Use the slope $- \frac{1}{6}$ and a given point $(4, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = - \frac{1}{6} \cdot (x - (4))$

Step 3

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

$y + 0 = - \frac{1}{6} \cdot (x - 4)$

Step 4

Solve for $y$ .

Add $y$ and $0$ .

$y = - \frac{1}{6} \cdot (x - 4)$

Simplify $- \frac{1}{6} \cdot (x - 4)$ .

Apply the distributive property.

$y = - \frac{1}{6} x - \frac{1}{6} \cdot - 4$

Combine $x$ and $\frac{1}{6}$ .

$y = - \frac{x}{6} - \frac{1}{6} \cdot - 4$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$y = - \frac{x}{6} + \frac{- 1}{6} \cdot - 4$

Factor $2$ out of $6$ .

$y = - \frac{x}{6} + \frac{- 1}{2 (3)} \cdot - 4$

Factor $2$ out of $- 4$ .

$y = - \frac{x}{6} + \frac{- 1}{2 \cdot 3} \cdot (2 \cdot - 2)$

Cancel the common factor.

$y = - \frac{x}{6} + \frac{- 1}{2 \cdot 3} \cdot (2 \cdot - 2)$

Rewrite the expression.

$y = - \frac{x}{6} + \frac{- 1}{3} \cdot - 2$

Combine $\frac{- 1}{3}$ and $- 2$ .

$y = - \frac{x}{6} + \frac{- - 2}{3}$

Multiply $- 1$ by $- 2$ .

$y = - \frac{x}{6} + \frac{2}{3}$

Step 5

Reorder terms.

$y = - (\frac{1}{6} x) + \frac{2}{3}$

Step 6

Remove parentheses.

$y = - \frac{1}{6} x + \frac{2}{3}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{1}{6} x + \frac{2}{3}$

Point-slope form:

$y + 0 = - \frac{1}{6} \cdot (x - 4)$

Step 8

Find the Equation Using Point-Slope Formula (4,0) , (-2,1)

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