Algebra Examples

$f (x) = \frac{10 x + 5}{3 - 7 x}$ , $g (x) = \frac{3 x - 5}{7 x + 10}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{10 x + 5}{3 - 7 x})$ by substituting in the value of $f$ into $g$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{3 (\frac{10 x + 5}{3 - 7 x}) - 5}{7 (\frac{10 x + 5}{3 - 7 x}) + 10}$

Step 3

Simplify with factoring out.

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Step 3.1

Factor $5$ out of $10 x + 5$ .

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Step 3.1.1

Factor $5$ out of $10 x$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{3 (\frac{5 (2 x) + 5}{3 - 7 x}) - 5}{7 (\frac{10 x + 5}{3 - 7 x}) + 10}$

Step 3.1.2

Factor $5$ out of $5$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{3 (\frac{5 (2 x) + 5 (1)}{3 - 7 x}) - 5}{7 (\frac{10 x + 5}{3 - 7 x}) + 10}$

Step 3.1.3

Factor $5$ out of $5 (2 x) + 5 (1)$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{3 (\frac{5 (2 x + 1)}{3 - 7 x}) - 5}{7 (\frac{10 x + 5}{3 - 7 x}) + 10}$

Step 3.2

Factor $5$ out of $10 x + 5$ .

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Step 3.2.1

Factor $5$ out of $10 x$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{3 (\frac{5 (2 x + 1)}{3 - 7 x}) - 5}{7 (\frac{5 (2 x) + 5}{3 - 7 x}) + 10}$

Step 3.2.2

Factor $5$ out of $5$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{3 (\frac{5 (2 x + 1)}{3 - 7 x}) - 5}{7 (\frac{5 (2 x) + 5 (1)}{3 - 7 x}) + 10}$

Step 3.2.3

Factor $5$ out of $5 (2 x) + 5 (1)$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{3 (\frac{5 (2 x + 1)}{3 - 7 x}) - 5}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $3 \frac{5 (2 x + 1)}{3 - 7 x}$ .

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Step 4.1.1

Combine $3$ and $\frac{5 (2 x + 1)}{3 - 7 x}$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{3 (5 (2 x + 1))}{3 - 7 x} - 5}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.1.2

Multiply $5$ by $3$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{15 (2 x + 1)}{3 - 7 x} - 5}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3 - 7 x}{3 - 7 x}$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{15 (2 x + 1)}{3 - 7 x} - 5 \cdot \frac{3 - 7 x}{3 - 7 x}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.3

Combine $- 5$ and $\frac{3 - 7 x}{3 - 7 x}$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{15 (2 x + 1)}{3 - 7 x} + \frac{- 5 (3 - 7 x)}{3 - 7 x}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.4

Combine the numerators over the common denominator.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{15 (2 x + 1) - 5 (3 - 7 x)}{3 - 7 x}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.5

Reorder terms.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{15 (2 x + 1) - 5 (3 - 7 x)}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6

Rewrite $\frac{15 (2 x + 1) - 5 (3 - 7 x)}{- 7 x + 3}$ in a factored form.

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Step 4.6.1

Factor $5$ out of $15 (2 x + 1) - 5 (3 - 7 x)$ .

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Step 4.6.1.1

Factor $5$ out of $15 (2 x + 1)$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (3 (2 x + 1)) - 5 (3 - 7 x)}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.1.2

Factor $5$ out of $- 5 (3 - 7 x)$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (3 (2 x + 1)) + 5 (- (3 - 7 x))}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.1.3

Factor $5$ out of $5 (3 (2 x + 1)) + 5 (- (3 - 7 x))$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (3 (2 x + 1) - (3 - 7 x))}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.2

Apply the distributive property.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (3 (2 x) + 3 \cdot 1 - (3 - 7 x))}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.3

Multiply $2$ by $3$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (6 x + 3 \cdot 1 - (3 - 7 x))}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.4

Multiply $3$ by $1$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (6 x + 3 - (3 - 7 x))}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.5

Apply the distributive property.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (6 x + 3 - 1 \cdot 3 - (- 7 x))}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.6

Multiply $- 1$ by $3$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (6 x + 3 - 3 - (- 7 x))}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.7

Multiply $- 7$ by $- 1$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (6 x + 3 - 3 + 7 x)}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.8

Add $6 x$ and $7 x$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (13 x + 3 - 3)}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.9

Subtract $3$ from $3$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 (13 x + 0)}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.10

Add $13 x$ and $0$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{5 \cdot (13 x)}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 4.6.11

Multiply $5$ by $13$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{7 (\frac{5 (2 x + 1)}{3 - 7 x}) + 10}$

Step 5

Simplify the denominator.

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Step 5.1

Multiply $7 \frac{5 (2 x + 1)}{3 - 7 x}$ .

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Step 5.1.1

Combine $7$ and $\frac{5 (2 x + 1)}{3 - 7 x}$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{7 (5 (2 x + 1))}{3 - 7 x} + 10}$

Step 5.1.2

Multiply $5$ by $7$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{35 (2 x + 1)}{3 - 7 x} + 10}$

Step 5.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{3 - 7 x}{3 - 7 x}$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{35 (2 x + 1)}{3 - 7 x} + \frac{10 (3 - 7 x)}{3 - 7 x}}$

Step 5.3

Combine the numerators over the common denominator.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{35 (2 x + 1) + 10 (3 - 7 x)}{3 - 7 x}}$

Step 5.4

Reorder terms.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{35 (2 x + 1) + 10 (3 - 7 x)}{- 7 x + 3}}$

Step 5.5

Rewrite $\frac{35 (2 x + 1) + 10 (3 - 7 x)}{- 7 x + 3}$ in a factored form.

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Step 5.5.1

Factor $5$ out of $35 (2 x + 1) + 10 (3 - 7 x)$ .

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Step 5.5.1.1

Factor $5$ out of $35 (2 x + 1)$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (7 (2 x + 1)) + 10 (3 - 7 x)}{- 7 x + 3}}$

Step 5.5.1.2

Factor $5$ out of $10 (3 - 7 x)$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (7 (2 x + 1)) + 5 (2 (3 - 7 x))}{- 7 x + 3}}$

Step 5.5.1.3

Factor $5$ out of $5 (7 (2 x + 1)) + 5 (2 (3 - 7 x))$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (7 (2 x + 1) + 2 (3 - 7 x))}{- 7 x + 3}}$

Step 5.5.2

Apply the distributive property.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (7 (2 x) + 7 \cdot 1 + 2 (3 - 7 x))}{- 7 x + 3}}$

Step 5.5.3

Multiply $2$ by $7$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (14 x + 7 \cdot 1 + 2 (3 - 7 x))}{- 7 x + 3}}$

Step 5.5.4

Multiply $7$ by $1$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (14 x + 7 + 2 (3 - 7 x))}{- 7 x + 3}}$

Step 5.5.5

Apply the distributive property.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (14 x + 7 + 2 \cdot 3 + 2 (- 7 x))}{- 7 x + 3}}$

Step 5.5.6

Multiply $2$ by $3$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (14 x + 7 + 6 + 2 (- 7 x))}{- 7 x + 3}}$

Step 5.5.7

Multiply $- 7$ by $2$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (14 x + 7 + 6 - 14 x)}{- 7 x + 3}}$

Step 5.5.8

Subtract $14 x$ from $14 x$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (0 + 7 + 6)}{- 7 x + 3}}$

Step 5.5.9

Add $0$ and $7$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 (7 + 6)}{- 7 x + 3}}$

Step 5.5.10

Add $7$ and $6$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{5 \cdot 13}{- 7 x + 3}}$

Step 5.6

Multiply $5$ by $13$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{\frac{65 x}{- 7 x + 3}}{\frac{65}{- 7 x + 3}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{65 x}{- 7 x + 3} \cdot \frac{- 7 x + 3}{65}$

Step 7

Cancel the common factor of $65$ .

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Step 7.1

Factor $65$ out of $65 x$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{65 (x)}{- 7 x + 3} \cdot \frac{- 7 x + 3}{65}$

Step 7.2

Cancel the common factor.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{65 x}{- 7 x + 3} \cdot \frac{- 7 x + 3}{65}$

Step 7.3

Rewrite the expression.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{x}{- 7 x + 3} \cdot (- 7 x + 3)$

Step 8

Multiply $\frac{x}{- 7 x + 3}$ by $- 7 x + 3$ .

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{x (- 7 x + 3)}{- 7 x + 3}$

Step 9

Cancel the common factor of $- 7 x + 3$ .

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Step 9.1

Cancel the common factor.

$g (\frac{10 x + 5}{3 - 7 x}) = \frac{x (- 7 x + 3)}{- 7 x + 3}$

Step 9.2

Divide $x$ by $1$ .

$g (\frac{10 x + 5}{3 - 7 x}) = x$