Algebra Examples

\frac{x}{2} + \frac{x}{3} = 5

$\frac{x}{2} + \frac{x}{3} = 5$

Step 1

Simplify

\frac{x}{2} + \frac{x}{3}

$\frac{x}{2} + \frac{x}{3}$ .

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Step 1.1

To write

\frac{x}{2}

$\frac{x}{2}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

\frac{x}{2} \cdot \frac{3}{3} + \frac{x}{3} = 5

$\frac{x}{2} \cdot \frac{3}{3} + \frac{x}{3} = 5$

Step 1.2

To write

\frac{x}{3}

$\frac{x}{3}$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

\frac{x}{2} \cdot \frac{3}{3} + \frac{x}{3} \cdot \frac{2}{2} = 5

$\frac{x}{2} \cdot \frac{3}{3} + \frac{x}{3} \cdot \frac{2}{2} = 5$

Step 1.3

Write each expression with a common denominator of

6

$6$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 1.3.1

Multiply

\frac{x}{2}

$\frac{x}{2}$ by

\frac{3}{3}

$\frac{3}{3}$ .

\frac{x \cdot 3}{2 \cdot 3} + \frac{x}{3} \cdot \frac{2}{2} = 5

$\frac{x \cdot 3}{2 \cdot 3} + \frac{x}{3} \cdot \frac{2}{2} = 5$

Step 1.3.2

Multiply

2

$2$ by

3

$3$ .

\frac{x \cdot 3}{6} + \frac{x}{3} \cdot \frac{2}{2} = 5

$\frac{x \cdot 3}{6} + \frac{x}{3} \cdot \frac{2}{2} = 5$

Step 1.3.3

Multiply

\frac{x}{3}

$\frac{x}{3}$ by

\frac{2}{2}

$\frac{2}{2}$ .

\frac{x \cdot 3}{6} + \frac{x \cdot 2}{3 \cdot 2} = 5

$\frac{x \cdot 3}{6} + \frac{x \cdot 2}{3 \cdot 2} = 5$

Step 1.3.4

Multiply

3

$3$ by

2

$2$ .

\frac{x \cdot 3}{6} + \frac{x \cdot 2}{6} = 5

$\frac{x \cdot 3}{6} + \frac{x \cdot 2}{6} = 5$

\frac{x \cdot 3}{6} + \frac{x \cdot 2}{6} = 5

$\frac{x \cdot 3}{6} + \frac{x \cdot 2}{6} = 5$

Step 1.4

Combine the numerators over the common denominator.

\frac{x \cdot 3 + x \cdot 2}{6} = 5

$\frac{x \cdot 3 + x \cdot 2}{6} = 5$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Move

3

$3$ to the left of

x

$x$ .

\frac{3 \cdot x + x \cdot 2}{6} = 5

$\frac{3 \cdot x + x \cdot 2}{6} = 5$

Step 1.5.2

Move

2

$2$ to the left of

x

$x$ .

\frac{3 x + 2 \cdot x}{6} = 5

$\frac{3 x + 2 \cdot x}{6} = 5$

Step 1.5.3

Add

3 x

$3 x$ and

2 x

$2 x$ .

\frac{5 x}{6} = 5

$\frac{5 x}{6} = 5$

\frac{5 x}{6} = 5

$\frac{5 x}{6} = 5$

\frac{5 x}{6} = 5

$\frac{5 x}{6} = 5$

Step 2

Multiply both sides of the equation by

\frac{6}{5}

$\frac{6}{5}$ .

\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 5

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 5$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify

\frac{6}{5} \cdot \frac{5 x}{6}

$\frac{6}{5} \cdot \frac{5 x}{6}$ .

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Step 3.1.1.1

Cancel the common factor of

6

$6$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 5$

Step 3.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot 5$

Step 3.1.1.2

Cancel the common factor of

$5$ .

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Step 3.1.1.2.1

Factor

$5$ out of

$5 x$ .

$\frac{1}{5} (5 (x)) = \frac{6}{5} \cdot 5$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot 5$

Step 3.1.1.2.3

Rewrite the expression.

$x = \frac{6}{5} \cdot 5$

Step 3.2

Simplify the right side.

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Step 3.2.1

Cancel the common factor of

$5$ .

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Step 3.2.1.1

Cancel the common factor.

$x = \frac{6}{5} \cdot 5$

Step 3.2.1.2

Rewrite the expression.

$x = 6$