Algebra Examples

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

Step 1

Simplify $\frac{a}{b} \cdot (1 - \frac{a}{x}) + \frac{b}{a} \cdot (1 - \frac{b}{x})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Simplify the numerator.

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Step 1.1.2.1

Write $1$ as a fraction with a common denominator.

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

Step 1.1.2.2

Combine the numerators over the common denominator.

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

Step 1.1.3

Combine $a$ and $\frac{x - a}{x}$ .

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

Step 1.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Write $1$ as a fraction with a common denominator.

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

Step 1.1.7.2

Combine the numerators over the common denominator.

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

Step 1.1.8

Combine $b$ and $\frac{x - b}{x}$ .

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

Step 1.1.9

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.10

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

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

Step 1.2

To write $\frac{a (x - a)}{x b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 1.3

To write $\frac{b (x - b)}{x a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

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

Step 1.4

Write each expression with a common denominator of $x b a$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{a (x - a)}{x b}$ by $\frac{a}{a}$ .

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

Step 1.4.2

Multiply $\frac{b (x - b)}{x a}$ by $\frac{b}{b}$ .

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

Step 1.4.3

Reorder the factors of $x b a$ .

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

Step 1.4.4

Reorder the factors of $x a b$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

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

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Step 1.6.1.1

Move $a$ .

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

Step 1.6.1.2

Multiply $a$ by $a$ .

$\frac{a^{2} (x - a) + b (x - b) b}{a b x} = 1$

Step 1.6.2

Apply the distributive property.

$\frac{a^{2} x + a^{2} (- a) + b (x - b) b}{a b x} = 1$

Step 1.6.3

Rewrite using the commutative property of multiplication.

$\frac{a^{2} x - a^{2} a + b (x - b) b}{a b x} = 1$

Step 1.6.4

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 1.6.4.1

Move $a$ .

$\frac{a^{2} x - (a \cdot a^{2}) + b (x - b) b}{a b x} = 1$

Step 1.6.4.2

Multiply $a$ by $a^{2}$ .

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Step 1.6.4.2.1

Raise $a$ to the power of $1$ .

$\frac{a^{2} x - (a^{1} a^{2}) + b (x - b) b}{a b x} = 1$

Step 1.6.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{a^{2} x - a^{1 + 2} + b (x - b) b}{a b x} = 1$

Step 1.6.4.3

Add $1$ and $2$ .

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

Step 1.6.5

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

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Step 1.6.5.1

Move $b$ .

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

Step 1.6.5.2

Multiply $b$ by $b$ .

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

Step 1.6.6

Apply the distributive property.

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

Step 1.6.7

Rewrite using the commutative property of multiplication.

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

Step 1.6.8

Multiply $b^{2}$ by $b$ by adding the exponents.

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Step 1.6.8.1

Move $b$ .

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

Step 1.6.8.2

Multiply $b$ by $b^{2}$ .

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Step 1.6.8.2.1

Raise $b$ to the power of $1$ .

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

Step 1.6.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.6.8.3

Add $1$ and $2$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$a b x, 1$

Step 2.2

The LCM of one and any expression is the expression.

$a b x$

Step 3

Multiply each term in $\frac{a^{2} x - a^{3} + b^{2} x - b^{3}}{a b x} = 1$ by $a b x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{a^{2} x - a^{3} + b^{2} x - b^{3}}{a b x} = 1$ by $a b x$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $a b x$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

$a^{2} x - a^{3} + b^{2} x - b^{3} = 1 (a b x)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $a b x$ by $1$ .

$a^{2} x - a^{3} + b^{2} x - b^{3} = a b x$

Step 4

Solve the equation.

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Step 4.1

Subtract $a b x$ from both sides of the equation.

$a^{2} x - a^{3} + b^{2} x - b^{3} - a b x = 0$

Step 4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 4.2.1

Add $a^{3}$ to both sides of the equation.

$a^{2} x + b^{2} x - b^{3} - a b x = a^{3}$

Step 4.2.2

Add $b^{3}$ to both sides of the equation.

$a^{2} x + b^{2} x - a b x = a^{3} + b^{3}$

Step 4.3

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

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Step 4.3.1

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

$x a^{2} + b^{2} x - a b x = a^{3} + b^{3}$

Step 4.3.2

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

$x a^{2} + x b^{2} - a b x = a^{3} + b^{3}$

Step 4.3.3

Factor $x$ out of $- a b x$ .

$x a^{2} + x b^{2} + x (- a b) = a^{3} + b^{3}$

Step 4.3.4

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

$x (a^{2} + b^{2}) + x (- a b) = a^{3} + b^{3}$

Step 4.3.5

Factor $x$ out of $x (a^{2} + b^{2}) + x (- a b)$ .

$x (a^{2} + b^{2} - a b) = a^{3} + b^{3}$

Step 4.4

Divide each term in $x (a^{2} + b^{2} - a b) = a^{3} + b^{3}$ by $a^{2} + b^{2} - a b$ and simplify.

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Step 4.4.1

Divide each term in $x (a^{2} + b^{2} - a b) = a^{3} + b^{3}$ by $a^{2} + b^{2} - a b$ .

$\frac{x (a^{2} + b^{2} - a b)}{a^{2} + b^{2} - a b} = \frac{a^{3}}{a^{2} + b^{2} - a b} + \frac{b^{3}}{a^{2} + b^{2} - a b}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $a^{2} + b^{2} - a b$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{x (a^{2} + b^{2} - a b)}{a^{2} + b^{2} - a b} = \frac{a^{3}}{a^{2} + b^{2} - a b} + \frac{b^{3}}{a^{2} + b^{2} - a b}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{a^{3}}{a^{2} + b^{2} - a b} + \frac{b^{3}}{a^{2} + b^{2} - a b}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Combine the numerators over the common denominator.

$x = \frac{a^{3} + b^{3}}{a^{2} + b^{2} - a b}$

Step 4.4.3.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = a$ and $b = b$ .

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

Step 4.4.3.3

Cancel the common factor of $a^{2} - a b + b^{2}$ and $a^{2} + b^{2} - a b$ .

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Step 4.4.3.3.1

Reorder terms.

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

Step 4.4.3.3.2

Cancel the common factor.

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

Step 4.4.3.3.3

Divide $a + b$ by $1$ .

$x = a + b$