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

$(0, 0)$ , $(5, 2)$

Step 1

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

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $0$ .

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

Add $2$ and $0$ .

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

Simplify the denominator.

Multiply $- 1$ by $0$ .

$m = \frac{2}{5 + 0}$

Add $5$ and $0$ .

$m = \frac{2}{5}$

Step 2

Use the slope $\frac{2}{5}$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = \frac{2}{5} \cdot (x - (0))$

Step 3

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

$y + 0 = \frac{2}{5} \cdot (x + 0)$

Step 4

Solve for $y$ .

Add $y$ and $0$ .

$y = \frac{2}{5} \cdot (x + 0)$

Simplify $\frac{2}{5} \cdot (x + 0)$ .

Add $x$ and $0$ .

$y = \frac{2}{5} \cdot x$

Combine $\frac{2}{5}$ and $x$ .

$y = \frac{2 x}{5}$

Step 5

Reorder terms.

$y = \frac{2}{5} x$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{2}{5} x$

Point-slope form:

$y + 0 = \frac{2}{5} \cdot (x + 0)$

Step 7

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

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