Precalculus Examples

$(a, \frac{1}{b})$ , $(b, \frac{1}{a})$

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.

$m = \frac{\frac{1}{a} - (\frac{1}{b})}{b - (a)}$

Step 4

Simplify.

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Step 4.1

Multiply the numerator and denominator of the fraction by $a b$ .

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Step 4.1.1

Multiply $\frac{\frac{1}{a} - \frac{1}{b}}{b - a}$ by $\frac{a b}{a b}$ .

$m = \frac{a b}{a b} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}$

Step 4.1.2

Combine.

$m = \frac{a b (\frac{1}{a} - \frac{1}{b})}{a b (b - a)}$

Step 4.2

Apply the distributive property.

$m = \frac{a b (\frac{1}{a}) + a b (- \frac{1}{b})}{a b \cdot b + a b (- a)}$

Step 4.3

Simplify by cancelling.

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Step 4.3.1

Cancel the common factor of $a$ .

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Step 4.3.1.1

Factor $a$ out of $a b$ .

$m = \frac{a (b) (\frac{1}{a}) + a b (- \frac{1}{b})}{a b \cdot b + a b (- a)}$

Step 4.3.1.2

Cancel the common factor.

$m = \frac{a b (\frac{1}{a}) + a b (- \frac{1}{b})}{a b \cdot b + a b (- a)}$

Step 4.3.1.3

Rewrite the expression.

$m = \frac{b + a b (- \frac{1}{b})}{a b \cdot b + a b (- a)}$

Step 4.3.2

Cancel the common factor of $b$ .

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Step 4.3.2.1

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

$m = \frac{b + a b (\frac{- 1}{b})}{a b \cdot b + a b (- a)}$

Step 4.3.2.2

Factor $b$ out of $a b$ .

$m = \frac{b + b a (\frac{- 1}{b})}{a b \cdot b + a b (- a)}$

Step 4.3.2.3

Cancel the common factor.

$m = \frac{b + b a (\frac{- 1}{b})}{a b \cdot b + a b (- a)}$

Step 4.3.2.4

Rewrite the expression.

$m = \frac{b + a \cdot - 1}{a b \cdot b + a b (- a)}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Move $- 1$ to the left of $a$ .

$m = \frac{b - 1 \cdot a}{a b \cdot b + a b (- a)}$

Step 4.4.2

Rewrite $- 1 a$ as $- a$ .

$m = \frac{b - a}{a b \cdot b + a b (- a)}$

Step 4.5

Simplify the denominator.

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Step 4.5.1

Multiply $b$ by $b$ by adding the exponents.

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Step 4.5.1.1

Move $b$ .

$m = \frac{b - a}{a (b \cdot b) + a b (- a)}$

Step 4.5.1.2

Multiply $b$ by $b$ .

$m = \frac{b - a}{a b^{2} + a b (- a)}$

Step 4.5.2

Rewrite using the commutative property of multiplication.

$m = \frac{b - a}{a b^{2} - a b a}$

Step 4.5.3

Multiply $a$ by $a$ by adding the exponents.

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Step 4.5.3.1

Move $a$ .

$m = \frac{b - a}{a b^{2} - a \cdot a b}$

Step 4.5.3.2

Multiply $a$ by $a$ .

$m = \frac{b - a}{a b^{2} - a^{2} b}$

Step 4.5.4

Factor $a b$ out of $a b^{2} - a^{2} b$ .

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Step 4.5.4.1

Factor $a b$ out of $a b^{2}$ .

$m = \frac{b - a}{a b (b) - a^{2} b}$

Step 4.5.4.2

Factor $a b$ out of $- a^{2} b$ .

$m = \frac{b - a}{a b (b) + a b (- a)}$

Step 4.5.4.3

Factor $a b$ out of $a b (b) + a b (- a)$ .

$m = \frac{b - a}{a b (b - a)}$

Step 4.6

Cancel the common factor of $b - a$ .

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Step 4.6.1

Cancel the common factor.

$m = \frac{b - a}{a b (b - a)}$

Step 4.6.2

Rewrite the expression.

$m = \frac{1}{a b}$