Precalculus Examples

$\frac{a x + b}{c x + d} = 9$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$c x + d, 1$

Step 1.2

Remove parentheses.

$c x + d, 1$

Step 1.3

The LCM of one and any expression is the expression.

$c x + d$

Step 2

Multiply each term in $\frac{a x + b}{c x + d} = 9$ by $c x + d$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{a x + b}{c x + d} = 9$ by $c x + d$ .

$\frac{a x + b}{c x + d} (c x + d) = 9 (c x + d)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $c x + d$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{a x + b}{c x + d} (c x + d) = 9 (c x + d)$

Step 2.2.1.2

Rewrite the expression.

$a x + b = 9 (c x + d)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Apply the distributive property.

$a x + b = 9 c x + 9 d$

Step 3

Solve the equation.

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Step 3.1

Subtract $9 c x$ from both sides of the equation.

$a x + b - 9 c x = 9 d$

Step 3.2

Subtract $b$ from both sides of the equation.

$a x - 9 c x = 9 d - b$

Step 3.3

Factor $x$ out of $a x - 9 c x$ .

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Step 3.3.1

Factor $x$ out of $a x$ .

$x a - 9 c x = 9 d - b$

Step 3.3.2

Factor $x$ out of $- 9 c x$ .

$x a + x (- 9 c) = 9 d - b$

Step 3.3.3

Factor $x$ out of $x a + x (- 9 c)$ .

$x (a - 9 c) = 9 d - b$

Step 3.4

Divide each term in $x (a - 9 c) = 9 d - b$ by $a - 9 c$ and simplify.

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Step 3.4.1

Divide each term in $x (a - 9 c) = 9 d - b$ by $a - 9 c$ .

$\frac{x (a - 9 c)}{a - 9 c} = \frac{9 d}{a - 9 c} + \frac{- b}{a - 9 c}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $a - 9 c$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{x (a - 9 c)}{a - 9 c} = \frac{9 d}{a - 9 c} + \frac{- b}{a - 9 c}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{9 d}{a - 9 c} + \frac{- b}{a - 9 c}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$x = \frac{9 d}{a - 9 c} - \frac{b}{a - 9 c}$

Step 3.4.3.2

Combine the numerators over the common denominator.

$x = \frac{9 d - b}{a - 9 c}$