Algebra Examples

$\frac{1}{2} + \frac{2}{x} = \frac{1}{3} + \frac{3}{x}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{1}{3}$ from both sides of the equation.

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

Step 1.2

Subtract $\frac{3}{x}$ from both sides of the equation.

$\frac{1}{2} + \frac{2}{x} - \frac{1}{3} - \frac{3}{x} = 0$

Step 2

Simplify $\frac{1}{2} + \frac{2}{x} - \frac{1}{3} - \frac{3}{x}$ .

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Step 2.1

Combine the numerators over the common denominator.

$\frac{1}{2} + \frac{2 - 3}{x} + \frac{- 1}{3} = 0$

Step 2.2

Subtract $3$ from $2$ .

$\frac{1}{2} + \frac{- 1}{x} + \frac{- 1}{3} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Move the negative in front of the fraction.

$\frac{1}{2} - \frac{1}{x} + \frac{- 1}{3} = 0$

Step 2.3.2

Move the negative in front of the fraction.

$\frac{1}{2} - \frac{1}{x} - \frac{1}{3} = 0$

Step 2.4

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

$- \frac{1}{x} + \frac{1}{2} \cdot \frac{3}{3} - \frac{1}{3} = 0$

Step 2.5

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

$- \frac{1}{x} + \frac{1}{2} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{2}{2} = 0$

Step 2.6

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$- \frac{1}{x} + \frac{3}{2 \cdot 3} - \frac{1}{3} \cdot \frac{2}{2} = 0$

Step 2.6.2

Multiply $2$ by $3$ .

$- \frac{1}{x} + \frac{3}{6} - \frac{1}{3} \cdot \frac{2}{2} = 0$

Step 2.6.3

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$- \frac{1}{x} + \frac{3}{6} - \frac{2}{3 \cdot 2} = 0$

Step 2.6.4

Multiply $3$ by $2$ .

$- \frac{1}{x} + \frac{3}{6} - \frac{2}{6} = 0$

Step 2.7

Combine the numerators over the common denominator.

$- \frac{1}{x} + \frac{3 - 2}{6} = 0$

Step 2.8

Subtract $2$ from $3$ .

$- \frac{1}{x} + \frac{1}{6} = 0$

Step 3

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

$- \frac{1}{x} \cdot \frac{6}{6} + \frac{1}{6} = 0$

Step 4

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

$- \frac{1}{x} \cdot \frac{6}{6} + \frac{1}{6} \cdot \frac{x}{x} = 0$

Step 5

Write each expression with a common denominator of $x \cdot 6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{1}{x}$ by $\frac{6}{6}$ .

$- \frac{6}{x \cdot 6} + \frac{1}{6} \cdot \frac{x}{x} = 0$

Step 5.2

Multiply $\frac{1}{6}$ by $\frac{x}{x}$ .

$- \frac{6}{x \cdot 6} + \frac{x}{6 x} = 0$

Step 5.3

Reorder the factors of $x \cdot 6$ .

$- \frac{6}{6 x} + \frac{x}{6 x} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- 6 + x}{6 x} = 0$

Step 7

Set the numerator equal to zero.

$- 6 + x = 0$

Step 8

Add $6$ to both sides of the equation.

$x = 6$