Algebra Examples

$\frac{\frac{1}{x} + \frac{1}{a}}{x + a} = 1$

Step 1

Factor each term.

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Step 1.1

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

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Step 1.1.1

Multiply $\frac{\frac{1}{x} + \frac{1}{a}}{x + a}$ by $\frac{x a}{x a}$ .

$\frac{x a}{x a} \cdot \frac{\frac{1}{x} + \frac{1}{a}}{x + a} = 1$

Step 1.1.2

Combine.

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

Step 1.2

Apply the distributive property.

$\frac{x a \frac{1}{x} + x a \frac{1}{a}}{x a x + x a \cdot a} = 1$

Step 1.3

Simplify by cancelling.

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Step 1.3.1

Cancel the common factor of $x$ .

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Step 1.3.1.1

Factor $x$ out of $x a$ .

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

Step 1.3.1.2

Cancel the common factor.

$\frac{x a \frac{1}{x} + x a \frac{1}{a}}{x a x + x a \cdot a} = 1$

Step 1.3.1.3

Rewrite the expression.

$\frac{a + x a \frac{1}{a}}{x a x + x a \cdot a} = 1$

Step 1.3.2

Cancel the common factor of $a$ .

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Step 1.3.2.1

Factor $a$ out of $x a$ .

$\frac{a + a x \frac{1}{a}}{x a x + x a \cdot a} = 1$

Step 1.3.2.2

Cancel the common factor.

$\frac{a + a x \frac{1}{a}}{x a x + x a \cdot a} = 1$

Step 1.3.2.3

Rewrite the expression.

$\frac{a + x}{x a x + x a \cdot a} = 1$

Step 1.4

Simplify the denominator.

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Step 1.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.1

Move $x$ .

$\frac{a + x}{x \cdot x a + x a \cdot a} = 1$

Step 1.4.1.2

Multiply $x$ by $x$ .

$\frac{a + x}{x^{2} a + x a \cdot a} = 1$

Step 1.4.2

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

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Step 1.4.2.1

Move $a$ .

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

Step 1.4.2.2

Multiply $a$ by $a$ .

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

Step 1.4.3

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

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Step 1.4.3.1

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

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

Step 1.4.3.2

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

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

Step 1.4.3.3

Factor $x a$ out of $x a (x) + x a (a)$ .

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

Step 1.5

Cancel the common factor of $a + x$ and $x + a$ .

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Step 1.5.1

Reorder terms.

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

$\frac{1}{x a} = 1$

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.

$x a, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x a$

Step 3

Multiply each term in $\frac{1}{x a} = 1$ by $x a$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{x a} = 1$ by $x a$ .

$\frac{1}{x a} (x a) = 1 (x a)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x a$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{1}{x a} (x a) = 1 (x a)$

Step 3.2.1.2

Rewrite the expression.

$1 = 1 (x a)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $x a$ by $1$ .

$1 = x a$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $x a = 1$ .

$x a = 1$

Step 4.2

Divide each term in $x a = 1$ by $a$ and simplify.

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Step 4.2.1

Divide each term in $x a = 1$ by $a$ .

$\frac{x a}{a} = \frac{1}{a}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $a$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{x a}{a} = \frac{1}{a}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{a}$