Algebra Examples

$\frac{2 x}{3} - \frac{5 x}{9} - \frac{x}{6} - \frac{5 x}{12} = \frac{13}{2}$

Step 1

Simplify $\frac{2 x}{3} - \frac{5 x}{9} - \frac{x}{6} - \frac{5 x}{12}$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

Multiply $\frac{2 x}{3}$ by $\frac{12}{12}$ .

$\frac{2 x}{3} \cdot \frac{12}{12} - \frac{5 x}{9} - \frac{x}{6} - \frac{5 x}{12} = \frac{13}{2}$

Step 1.1.2

Multiply $\frac{2 x}{3}$ by $\frac{12}{12}$ .

$\frac{2 x \cdot 12}{3 \cdot 12} - \frac{5 x}{9} - \frac{x}{6} - \frac{5 x}{12} = \frac{13}{2}$

Step 1.1.3

Multiply $\frac{5 x}{9}$ by $\frac{4}{4}$ .

$\frac{2 x \cdot 12}{3 \cdot 12} - (\frac{5 x}{9} \cdot \frac{4}{4}) - \frac{x}{6} - \frac{5 x}{12} = \frac{13}{2}$

Step 1.1.4

Multiply $\frac{5 x}{9}$ by $\frac{4}{4}$ .

$\frac{2 x \cdot 12}{3 \cdot 12} - \frac{5 x \cdot 4}{9 \cdot 4} - \frac{x}{6} - \frac{5 x}{12} = \frac{13}{2}$

Step 1.1.5

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

$\frac{2 x \cdot 12}{3 \cdot 12} - \frac{5 x \cdot 4}{9 \cdot 4} - (\frac{x}{6} \cdot \frac{6}{6}) - \frac{5 x}{12} = \frac{13}{2}$

Step 1.1.6

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

$\frac{2 x \cdot 12}{3 \cdot 12} - \frac{5 x \cdot 4}{9 \cdot 4} - \frac{x \cdot 6}{6 \cdot 6} - \frac{5 x}{12} = \frac{13}{2}$

Step 1.1.7

Multiply $\frac{5 x}{12}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 12}{3 \cdot 12} - \frac{5 x \cdot 4}{9 \cdot 4} - \frac{x \cdot 6}{6 \cdot 6} - (\frac{5 x}{12} \cdot \frac{3}{3}) = \frac{13}{2}$

Step 1.1.8

Multiply $\frac{5 x}{12}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 12}{3 \cdot 12} - \frac{5 x \cdot 4}{9 \cdot 4} - \frac{x \cdot 6}{6 \cdot 6} - \frac{5 x \cdot 3}{12 \cdot 3} = \frac{13}{2}$

Step 1.1.9

Multiply $3$ by $12$ .

$\frac{2 x \cdot 12}{36} - \frac{5 x \cdot 4}{9 \cdot 4} - \frac{x \cdot 6}{6 \cdot 6} - \frac{5 x \cdot 3}{12 \cdot 3} = \frac{13}{2}$

Step 1.1.10

Reorder the factors of $9 \cdot 4$ .

$\frac{2 x \cdot 12}{36} - \frac{5 x \cdot 4}{4 \cdot 9} - \frac{x \cdot 6}{6 \cdot 6} - \frac{5 x \cdot 3}{12 \cdot 3} = \frac{13}{2}$

Step 1.1.11

Multiply $4$ by $9$ .

$\frac{2 x \cdot 12}{36} - \frac{5 x \cdot 4}{36} - \frac{x \cdot 6}{6 \cdot 6} - \frac{5 x \cdot 3}{12 \cdot 3} = \frac{13}{2}$

Step 1.1.12

Multiply $6$ by $6$ .

$\frac{2 x \cdot 12}{36} - \frac{5 x \cdot 4}{36} - \frac{x \cdot 6}{36} - \frac{5 x \cdot 3}{12 \cdot 3} = \frac{13}{2}$

Step 1.1.13

Reorder the factors of $12 \cdot 3$ .

$\frac{2 x \cdot 12}{36} - \frac{5 x \cdot 4}{36} - \frac{x \cdot 6}{36} - \frac{5 x \cdot 3}{3 \cdot 12} = \frac{13}{2}$

Step 1.1.14

Multiply $3$ by $12$ .

$\frac{2 x \cdot 12}{36} - \frac{5 x \cdot 4}{36} - \frac{x \cdot 6}{36} - \frac{5 x \cdot 3}{36} = \frac{13}{2}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{2 x \cdot 12 - 5 x \cdot 4 - x \cdot 6 - 5 x \cdot 3}{36} = \frac{13}{2}$

Step 1.3

Simplify each term.

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Step 1.3.1

Multiply $12$ by $2$ .

$\frac{24 x - 5 x \cdot 4 - x \cdot 6 - 5 x \cdot 3}{36} = \frac{13}{2}$

Step 1.3.2

Multiply $4$ by $- 5$ .

$\frac{24 x - 20 x - x \cdot 6 - 5 x \cdot 3}{36} = \frac{13}{2}$

Step 1.3.3

Multiply $6$ by $- 1$ .

$\frac{24 x - 20 x - 6 x - 5 x \cdot 3}{36} = \frac{13}{2}$

Step 1.3.4

Multiply $3$ by $- 5$ .

$\frac{24 x - 20 x - 6 x - 15 x}{36} = \frac{13}{2}$

Step 1.4

Simplify by adding terms.

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Step 1.4.1

Subtract $20 x$ from $24 x$ .

$\frac{4 x - 6 x - 15 x}{36} = \frac{13}{2}$

Step 1.4.2

Subtract $6 x$ from $4 x$ .

$\frac{- 2 x - 15 x}{36} = \frac{13}{2}$

Step 1.4.3

Subtract $15 x$ from $- 2 x$ .

$\frac{- 17 x}{36} = \frac{13}{2}$

Step 1.4.4

Move the negative in front of the fraction.

$- \frac{17 x}{36} = \frac{13}{2}$

Step 2

Multiply both sides of the equation by $- \frac{36}{17}$ .

$- \frac{36}{17} (- \frac{17 x}{36}) = - \frac{36}{17} \cdot \frac{13}{2}$

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 $- \frac{36}{17} (- \frac{17 x}{36})$ .

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Step 3.1.1.1

Cancel the common factor of $36$ .

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Step 3.1.1.1.1

Move the leading negative in $- \frac{36}{17}$ into the numerator.

$\frac{- 36}{17} (- \frac{17 x}{36}) = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.1.2

Move the leading negative in $- \frac{17 x}{36}$ into the numerator.

$\frac{- 36}{17} \cdot \frac{- 17 x}{36} = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.1.3

Factor $36$ out of $- 36$ .

$\frac{36 (- 1)}{17} \cdot \frac{- 17 x}{36} = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.1.4

Cancel the common factor.

$\frac{36 \cdot - 1}{17} \cdot \frac{- 17 x}{36} = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{17} (- 17 x) = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.2

Cancel the common factor of $17$ .

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Step 3.1.1.2.1

Factor $17$ out of $- 17 x$ .

$\frac{- 1}{17} (17 (- x)) = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{17} (17 (- x)) = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.2.3

Rewrite the expression.

$- - x = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.3

Multiply.

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Step 3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{36}{17} \cdot \frac{13}{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{36}{17} \cdot \frac{13}{2}$ .

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Step 3.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1

Move the leading negative in $- \frac{36}{17}$ into the numerator.

$x = \frac{- 36}{17} \cdot \frac{13}{2}$

Step 3.2.1.1.2

Factor $2$ out of $- 36$ .

$x = \frac{2 (- 18)}{17} \cdot \frac{13}{2}$

Step 3.2.1.1.3

Cancel the common factor.

$x = \frac{2 \cdot - 18}{17} \cdot \frac{13}{2}$

Step 3.2.1.1.4

Rewrite the expression.

$x = \frac{- 18}{17} \cdot 13$

Step 3.2.1.2

Combine $\frac{- 18}{17}$ and $13$ .

$x = \frac{- 18 \cdot 13}{17}$

Step 3.2.1.3

Simplify the expression.

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Step 3.2.1.3.1

Multiply $- 18$ by $13$ .

$x = \frac{- 234}{17}$

Step 3.2.1.3.2

Move the negative in front of the fraction.

$x = - \frac{234}{17}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{234}{17}$

Decimal Form:

$x = - 13.76470588 \dots$

Mixed Number Form:

$x = - 13 \frac{13}{17}$