Precalculus Examples

$(\frac{2}{3}, 0)$ , $(0, - 2)$

Step 1

Find the slope of the line between $(\frac{2}{3}, 0)$ and $(0, - 2)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

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).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

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

Step 1.4

Simplify.

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Step 1.4.1

Cancel the common factor of $- 2 - (0)$ and $0 - (\frac{2}{3})$ .

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Step 1.4.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$m = \frac{- 1 \cdot 2 - (0)}{0 - (\frac{2}{3})}$

Step 1.4.1.2

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

$m = \frac{- 1 (2 + 0)}{0 - (\frac{2}{3})}$

Step 1.4.1.3

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

$m = \frac{2 (- 1 (1 + 0))}{0 - \frac{2}{3}}$

Step 1.4.1.4

Cancel the common factors.

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Step 1.4.1.4.1

Factor $2$ out of $0$ .

$m = \frac{2 (- 1 (1 + 0))}{2 (0) - \frac{2}{3}}$

Step 1.4.1.4.2

Factor $2$ out of $- \frac{2}{3}$ .

$m = \frac{2 (- 1 (1 + 0))}{2 (0) + 2 (- \frac{1}{3})}$

Step 1.4.1.4.3

Factor $2$ out of $2 (0) + 2 (- \frac{1}{3})$ .

$m = \frac{2 (- 1 (1 + 0))}{2 (0 - \frac{1}{3})}$

Step 1.4.1.4.4

Cancel the common factor.

$m = \frac{2 (- 1 (1 + 0))}{2 (0 - \frac{1}{3})}$

Step 1.4.1.4.5

Rewrite the expression.

$m = \frac{- 1 (1 + 0)}{0 - \frac{1}{3}}$

Step 1.4.2

Simplify the expression.

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Step 1.4.2.1

Add $1$ and $0$ .

$m = \frac{- 1 \cdot 1}{0 - \frac{1}{3}}$

Step 1.4.2.2

Subtract $\frac{1}{3}$ from $0$ .

$m = \frac{- 1 \cdot 1}{- \frac{1}{3}}$

Step 1.4.2.3

Multiply $- 1$ by $1$ .

$m = \frac{- 1}{- \frac{1}{3}}$

Step 1.4.3

Dividing two negative values results in a positive value.

$m = \frac{1}{\frac{1}{3}}$

Step 1.4.4

Multiply the numerator by the reciprocal of the denominator.

$m = 1 \cdot 3$

Step 1.4.5

Multiply $3$ by $1$ .

$m = 3$

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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Step 4.1

Add $y$ and $0$ .

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

$y = 3 x - 2$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 3 x - 2$

Point-slope form:

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

Step 6