Precalculus Examples

$(\frac{5}{6}, - \frac{1}{2})$ and $(\frac{1}{3}, \frac{1}{3})$

Step 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 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 3

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

$m = \frac{\frac{1}{3} - (- \frac{1}{2})}{\frac{1}{3} - (\frac{5}{6})}$

Step 4

Simplify.

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Step 4.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 4.1.1

Multiply $\frac{\frac{1}{3} - - \frac{1}{2}}{\frac{1}{3} - \frac{5}{6}}$ by $\frac{6}{6}$ .

$m = \frac{6}{6} \cdot \frac{\frac{1}{3} + \frac{1}{2}}{\frac{1}{3} - \frac{5}{6}}$

Step 4.1.2

Combine.

$m = \frac{6 (\frac{1}{3} + \frac{1}{2})}{6 (\frac{1}{3} - \frac{5}{6})}$

Step 4.2

Apply the distributive property.

$m = \frac{6 (\frac{1}{3}) + 6 (\frac{1}{2})}{6 (\frac{1}{3}) + 6 (- \frac{5}{6})}$

Step 4.3

Simplify by cancelling.

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Step 4.3.1

Cancel the common factor of $3$ .

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Step 4.3.1.1

Factor $3$ out of $6$ .

$m = \frac{3 (2) (\frac{1}{3}) + 6 (\frac{1}{2})}{6 (\frac{1}{3}) + 6 (- \frac{5}{6})}$

Step 4.3.1.2

Cancel the common factor.

$m = \frac{3 \cdot (2 (\frac{1}{3})) + 6 (\frac{1}{2})}{6 (\frac{1}{3}) + 6 (- \frac{5}{6})}$

Step 4.3.1.3

Rewrite the expression.

$m = \frac{2 + 6 (\frac{1}{2})}{6 (\frac{1}{3}) + 6 (- \frac{5}{6})}$

Step 4.3.2

Cancel the common factor of $3$ .

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Step 4.3.2.1

Factor $3$ out of $6$ .

$m = \frac{2 + 6 (\frac{1}{2})}{3 (2) (\frac{1}{3}) + 6 (- \frac{5}{6})}$

Step 4.3.2.2

Cancel the common factor.

$m = \frac{2 + 6 (\frac{1}{2})}{3 \cdot (2 (\frac{1}{3})) + 6 (- \frac{5}{6})}$

Step 4.3.2.3

Rewrite the expression.

$m = \frac{2 + 6 (\frac{1}{2})}{2 + 6 (- \frac{5}{6})}$

Step 4.3.3

Cancel the common factor of $6$ .

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Step 4.3.3.1

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$m = \frac{2 + 6 (\frac{1}{2})}{2 + 6 (\frac{- 5}{6})}$

Step 4.3.3.2

Cancel the common factor.

$m = \frac{2 + 6 (\frac{1}{2})}{2 + 6 (\frac{- 5}{6})}$

Step 4.3.3.3

Rewrite the expression.

$m = \frac{2 + 6 (\frac{1}{2})}{2 - 5}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Multiply $- - \frac{1}{2}$ .

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Step 4.4.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{2 + 6 (1 (\frac{1}{2}))}{2 - 5}$

Step 4.4.1.2

Multiply $\frac{1}{2}$ by $1$ .

$m = \frac{2 + 6 (\frac{1}{2})}{2 - 5}$

Step 4.4.2

Cancel the common factor of $2$ .

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Step 4.4.2.1

Factor $2$ out of $6$ .

$m = \frac{2 + 2 (3) (\frac{1}{2})}{2 - 5}$

Step 4.4.2.2

Cancel the common factor.

$m = \frac{2 + 2 \cdot (3 (\frac{1}{2}))}{2 - 5}$

Step 4.4.2.3

Rewrite the expression.

$m = \frac{2 + 3}{2 - 5}$

Step 4.4.3

Add $2$ and $3$ .

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

Step 4.5

Simplify the expression.

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Step 4.5.1

Subtract $5$ from $2$ .

$m = \frac{5}{- 3}$

Step 4.5.2

Move the negative in front of the fraction.

$m = - \frac{5}{3}$

Step 5