Algebra Examples

$\frac{x^{2}}{3 a x - 2 - x + 6 a} - \frac{x}{3 a - 1}$

Step 1

To write $\frac{x^{2}}{3 a x - 2 - x + 6 a}$ as a fraction with a common denominator, multiply by $\frac{3 a - 1}{3 a - 1}$ .

$\frac{x^{2}}{3 a x - 2 - x + 6 a} \cdot \frac{3 a - 1}{3 a - 1} - \frac{x}{3 a - 1}$

Step 2

To write $- \frac{x}{3 a - 1}$ as a fraction with a common denominator, multiply by $\frac{3 a x - 2 - x + 6 a}{3 a x - 2 - x + 6 a}$ .

$\frac{x^{2}}{3 a x - 2 - x + 6 a} \cdot \frac{3 a - 1}{3 a - 1} - \frac{x}{3 a - 1} \cdot \frac{3 a x - 2 - x + 6 a}{3 a x - 2 - x + 6 a}$

Step 3

Write each expression with a common denominator of $(3 a x - 2 - x + 6 a) (3 a - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x^{2}}{3 a x - 2 - x + 6 a}$ by $\frac{3 a - 1}{3 a - 1}$ .

$\frac{x^{2} (3 a - 1)}{(3 a x - 2 - x + 6 a) (3 a - 1)} - \frac{x}{3 a - 1} \cdot \frac{3 a x - 2 - x + 6 a}{3 a x - 2 - x + 6 a}$

Step 3.2

Multiply $\frac{x}{3 a - 1}$ by $\frac{3 a x - 2 - x + 6 a}{3 a x - 2 - x + 6 a}$ .

$\frac{x^{2} (3 a - 1)}{(3 a x - 2 - x + 6 a) (3 a - 1)} - \frac{x (3 a x - 2 - x + 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 3.3

Reorder the factors of $(3 a x - 2 - x + 6 a) (3 a - 1)$ .

$\frac{x^{2} (3 a - 1)}{(3 a - 1) (3 a x - 2 - x + 6 a)} - \frac{x (3 a x - 2 - x + 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x^{2} (3 a - 1) - x (3 a x - 2 - x + 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $x$ out of $x^{2} (3 a - 1) - x (3 a x - 2 - x + 6 a)$ .

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Step 5.1.1

Factor $x$ out of $x^{2} (3 a - 1)$ .

$\frac{x (x (3 a - 1)) - x (3 a x - 2 - x + 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.1.2

Factor $x$ out of $- x (3 a x - 2 - x + 6 a)$ .

$\frac{x (x (3 a - 1)) + x (- (3 a x - 2 - x + 6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.1.3

Factor $x$ out of $x (x (3 a - 1)) + x (- (3 a x - 2 - x + 6 a))$ .

$\frac{x (x (3 a - 1) - (3 a x - 2 - x + 6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.2

Apply the distributive property.

$\frac{x (x (3 a) + x \cdot - 1 - (3 a x - 2 - x + 6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.3

Rewrite using the commutative property of multiplication.

$\frac{x (3 x a + x \cdot - 1 - (3 a x - 2 - x + 6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.4

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

$\frac{x (3 x a - 1 \cdot x - (3 a x - 2 - x + 6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.5

Rewrite $- 1 x$ as $- x$ .

$\frac{x (3 x a - x - (3 a x - 2 - x + 6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.6

Apply the distributive property.

$\frac{x (3 x a - x - (3 a x) - - 2 - - x - (6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.7

Simplify.

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Step 5.7.1

Multiply $3$ by $- 1$ .

$\frac{x (3 x a - x - 3 (a x) - - 2 - - x - (6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.7.2

Multiply $- 1$ by $- 2$ .

$\frac{x (3 x a - x - 3 (a x) + 2 - - x - (6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.7.3

Multiply $- - x$ .

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Step 5.7.3.1

Multiply $- 1$ by $- 1$ .

$\frac{x (3 x a - x - 3 (a x) + 2 + 1 x - (6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.7.3.2

Multiply $x$ by $1$ .

$\frac{x (3 x a - x - 3 (a x) + 2 + x - (6 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.7.4

Multiply $6$ by $- 1$ .

$\frac{x (3 x a - x - 3 (a x) + 2 + x - 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.8

Subtract $3 a x$ from $3 x a$ .

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Step 5.8.1

Move $x$ .

$\frac{x (- x + 3 a x - 3 a x + 2 + x - 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.8.2

Subtract $3 a x$ from $3 a x$ .

$\frac{x (- x + 0 + 2 + x - 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.9

Add $- x$ and $0$ .

$\frac{x (- x + 2 + x - 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.10

Add $- x$ and $x$ .

$\frac{x (0 + 2 - 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.11

Add $0$ and $2$ .

$\frac{x (2 - 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.12

Factor $2$ out of $2 - 6 a$ .

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Step 5.12.1

Factor $2$ out of $2$ .

$\frac{x (2 (1) - 6 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.12.2

Factor $2$ out of $- 6 a$ .

$\frac{x (2 (1) + 2 (- 3 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 5.12.3

Factor $2$ out of $2 (1) + 2 (- 3 a)$ .

$\frac{x (2 (1 - 3 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

$\frac{x \cdot 2 (1 - 3 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $1 - 3 a$ and $3 a - 1$ .

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Step 6.1.1

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

$\frac{x \cdot 2 (- 1 (- 1) - 3 a)}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 6.1.2

Factor $- 1$ out of $- 3 a$ .

$\frac{x \cdot 2 (- 1 (- 1) - (3 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 6.1.3

Factor $- 1$ out of $- 1 (- 1) - (3 a)$ .

$\frac{x \cdot 2 (- 1 (- 1 + 3 a))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 6.1.4

Reorder terms.

$\frac{x \cdot 2 (- 1 (3 a - 1))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 6.1.5

Cancel the common factor.

$\frac{x \cdot 2 (- 1 (3 a - 1))}{(3 a - 1) (3 a x - 2 - x + 6 a)}$

Step 6.1.6

Rewrite the expression.

$\frac{x \cdot 2 \cdot (- 1)}{3 a x - 2 - x + 6 a}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Multiply $- 1$ by $2$ .

$\frac{x \cdot - 2}{3 a x - 2 - x + 6 a}$

Step 6.2.2

Move $- 2$ to the left of $x$ .

$\frac{- 2 \cdot x}{3 a x - 2 - x + 6 a}$

Step 6.2.3

Move the negative in front of the fraction.

$- \frac{2 x}{3 a x - 2 - x + 6 a}$