Basic Math Examples

$\frac{1}{3} a - 1 = - \frac{1}{2} a - a$

Step 1

Since $a$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \frac{1}{2} a - a = \frac{1}{3} a - 1$

Step 2

Simplify $- \frac{1}{2} a - a$ .

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Step 2.1

Combine $a$ and $\frac{1}{2}$ .

$- \frac{a}{2} - a = \frac{1}{3} a - 1$

Step 2.2

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

$- \frac{a}{2} - a \cdot \frac{2}{2} = \frac{1}{3} a - 1$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $- a$ and $\frac{2}{2}$ .

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

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{- a - a \cdot 2}{2} = \frac{1}{3} a - 1$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Factor $a$ out of $- a - a \cdot 2$ .

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Step 2.4.1.1

Factor $a$ out of $- a$ .

$\frac{a \cdot - 1 - a \cdot 2}{2} = \frac{1}{3} a - 1$

Step 2.4.1.2

Factor $a$ out of $- a \cdot 2$ .

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

Step 2.4.1.3

Factor $a$ out of $a \cdot - 1 + a (- 1 \cdot 2)$ .

$\frac{a (- 1 - 1 \cdot 2)}{2} = \frac{1}{3} a - 1$

Step 2.4.2

Multiply $- 1$ by $2$ .

$\frac{a (- 1 - 2)}{2} = \frac{1}{3} a - 1$

Step 2.4.3

Subtract $2$ from $- 1$ .

$\frac{a \cdot - 3}{2} = \frac{1}{3} a - 1$

Step 2.5

Simplify the expression.

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Step 2.5.1

Move $- 3$ to the left of $a$ .

$\frac{- 3 \cdot a}{2} = \frac{1}{3} a - 1$

Step 2.5.2

Move the negative in front of the fraction.

$- \frac{3 a}{2} = \frac{1}{3} a - 1$

Step 3

Combine $\frac{1}{3}$ and $a$ .

$- \frac{3 a}{2} = \frac{a}{3} - 1$

Step 4

Move all terms containing $a$ to the left side of the equation.

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Step 4.1

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

$- \frac{3 a}{2} - \frac{a}{3} = - 1$

Step 4.2

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

$- \frac{3 a}{2} \cdot \frac{3}{3} - \frac{a}{3} = - 1$

Step 4.3

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

$- \frac{3 a}{2} \cdot \frac{3}{3} - \frac{a}{3} \cdot \frac{2}{2} = - 1$

Step 4.4

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

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Step 4.4.1

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

$- \frac{3 a \cdot 3}{2 \cdot 3} - \frac{a}{3} \cdot \frac{2}{2} = - 1$

Step 4.4.2

Multiply $2$ by $3$ .

$- \frac{3 a \cdot 3}{6} - \frac{a}{3} \cdot \frac{2}{2} = - 1$

Step 4.4.3

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

$- \frac{3 a \cdot 3}{6} - \frac{a \cdot 2}{3 \cdot 2} = - 1$

Step 4.4.4

Multiply $3$ by $2$ .

$- \frac{3 a \cdot 3}{6} - \frac{a \cdot 2}{6} = - 1$

Step 4.5

Combine the numerators over the common denominator.

$\frac{- 3 a \cdot 3 - a \cdot 2}{6} = - 1$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Factor $a$ out of $- 3 a \cdot 3 - a \cdot 2$ .

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Step 4.6.1.1

Factor $a$ out of $- 3 a \cdot 3$ .

$\frac{a (- 3 \cdot 3) - a \cdot 2}{6} = - 1$

Step 4.6.1.2

Factor $a$ out of $- a \cdot 2$ .

$\frac{a (- 3 \cdot 3) + a (- 1 \cdot 2)}{6} = - 1$

Step 4.6.1.3

Factor $a$ out of $a (- 3 \cdot 3) + a (- 1 \cdot 2)$ .

$\frac{a (- 3 \cdot 3 - 1 \cdot 2)}{6} = - 1$

Step 4.6.2

Multiply $- 3$ by $3$ .

$\frac{a (- 9 - 1 \cdot 2)}{6} = - 1$

Step 4.6.3

Multiply $- 1$ by $2$ .

$\frac{a (- 9 - 2)}{6} = - 1$

Step 4.6.4

Subtract $2$ from $- 9$ .

$\frac{a \cdot - 11}{6} = - 1$

Step 4.7

Move $- 11$ to the left of $a$ .

$\frac{- 11 \cdot a}{6} = - 1$

Step 4.8

Move the negative in front of the fraction.

$- \frac{11 a}{6} = - 1$

Step 5

Multiply both sides of the equation by $- \frac{6}{11}$ .

$- \frac{6}{11} (- \frac{11 a}{6}) = - \frac{6}{11} \cdot - 1$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $- \frac{6}{11} (- \frac{11 a}{6})$ .

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Step 6.1.1.1

Cancel the common factor of $6$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{6}{11}$ into the numerator.

$\frac{- 6}{11} (- \frac{11 a}{6}) = - \frac{6}{11} \cdot - 1$

Step 6.1.1.1.2

Move the leading negative in $- \frac{11 a}{6}$ into the numerator.

$\frac{- 6}{11} \cdot \frac{- 11 a}{6} = - \frac{6}{11} \cdot - 1$

Step 6.1.1.1.3

Factor $6$ out of $- 6$ .

$\frac{6 (- 1)}{11} \cdot \frac{- 11 a}{6} = - \frac{6}{11} \cdot - 1$

Step 6.1.1.1.4

Cancel the common factor.

$\frac{6 \cdot - 1}{11} \cdot \frac{- 11 a}{6} = - \frac{6}{11} \cdot - 1$

Step 6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{11} (- 11 a) = - \frac{6}{11} \cdot - 1$

Step 6.1.1.2

Cancel the common factor of $11$ .

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Step 6.1.1.2.1

Factor $11$ out of $- 11 a$ .

$\frac{- 1}{11} (11 (- a)) = - \frac{6}{11} \cdot - 1$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{11} (11 (- a)) = - \frac{6}{11} \cdot - 1$

Step 6.1.1.2.3

Rewrite the expression.

$- - a = - \frac{6}{11} \cdot - 1$

Step 6.1.1.3

Multiply.

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Step 6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 a = - \frac{6}{11} \cdot - 1$

Step 6.1.1.3.2

Multiply $a$ by $1$ .

$a = - \frac{6}{11} \cdot - 1$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- \frac{6}{11} \cdot - 1$ .

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Step 6.2.1.1

Multiply $- 1$ by $- 1$ .

$a = 1 (\frac{6}{11})$

Step 6.2.1.2

Multiply $\frac{6}{11}$ by $1$ .

$a = \frac{6}{11}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{6}{11}$

Decimal Form:

$a = 0.\overline{54}$