Find the Equation Using Point-Slope Formula (0,6) , (3,4)

$(0, 6)$ , $(3, 4)$

Step 1

Find the slope of the line between $(0, 6)$ and $(3, 4)$ 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.

$m = \frac{4 - (6)}{3 - (0)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $6$ .

$m = \frac{4 - 6}{3 - (0)}$

Subtract $6$ from $4$ .

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

Simplify the denominator.

Multiply $- 1$ by $0$ .

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

Add $3$ and $0$ .

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

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{2}{3}$ and a given point $(0, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = - \frac{2}{3} \cdot (x - (0))$

Step 3

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

$y - 6 = - \frac{2}{3} \cdot (x + 0)$

Step 4

Solve for $y$ .

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

Add $x$ and $0$ .

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

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

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

Move $2$ to the left of $x$ .

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

Add $6$ to both sides of the equation.

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

Step 5

Reorder terms.

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

Step 6

Remove parentheses.

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

Step 7

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

$y - 6 = - \frac{2}{3} \cdot (x + 0)$

Step 8

Find the Equation Using Point-Slope Formula (0,6) , (3,4)

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