Basic Math Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Move the negative in front of the fraction.

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

Step 1.1.5

Apply the distributive property.

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

Step 1.1.6

Multiply $- 1$ by $1$ .

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

Step 1.1.7

Multiply $- \frac{1}{3} (- a)$ .

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Step 1.1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.7.2

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

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

Step 1.1.7.3

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Multiply $2$ by $3$ .

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $3$ by $2$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Reorder $a$ and $2$ .

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

Step 1.7

Add $3 \cdot a$ and $2 \cdot a$ .

$\frac{5 \cdot a}{6} - \frac{1}{2} - \frac{1}{3} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.8

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

$\frac{5 a}{6} - \frac{1}{2} \cdot \frac{3}{3} - \frac{1}{3} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.9

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

$\frac{5 a}{6} - \frac{1}{2} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{2}{2} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.10

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

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Step 1.10.1

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

$\frac{5 a}{6} - \frac{3}{2 \cdot 3} - \frac{1}{3} \cdot \frac{2}{2} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.10.2

Multiply $2$ by $3$ .

$\frac{5 a}{6} - \frac{3}{6} - \frac{1}{3} \cdot \frac{2}{2} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.10.3

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

$\frac{5 a}{6} - \frac{3}{6} - \frac{2}{3 \cdot 2} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.10.4

Multiply $3$ by $2$ .

$\frac{5 a}{6} - \frac{3}{6} - \frac{2}{6} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.11

Combine the numerators over the common denominator.

$\frac{5 a}{6} + \frac{- 3 - 2}{6} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.12

Subtract $2$ from $- 3$ .

$\frac{5 a}{6} + \frac{- 5}{6} = a + \frac{1}{6} \cdot (a - 1)$

Step 1.13

Move the negative in front of the fraction.

$\frac{5 a}{6} - \frac{5}{6} = a + \frac{1}{6} \cdot (a - 1)$

Step 2

Simplify $a + \frac{1}{6} \cdot (a - 1)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{5 a}{6} - \frac{5}{6} = a + \frac{1}{6} a + \frac{1}{6} \cdot - 1$

Step 2.1.2

Combine $\frac{1}{6}$ and $a$ .

$\frac{5 a}{6} - \frac{5}{6} = a + \frac{a}{6} + \frac{1}{6} \cdot - 1$

Step 2.1.3

Combine $\frac{1}{6}$ and $- 1$ .

$\frac{5 a}{6} - \frac{5}{6} = a + \frac{a}{6} + \frac{- 1}{6}$

Step 2.1.4

Move the negative in front of the fraction.

$\frac{5 a}{6} - \frac{5}{6} = a + \frac{a}{6} - \frac{1}{6}$

Step 2.2

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

$\frac{5 a}{6} - \frac{5}{6} = a \cdot \frac{6}{6} + \frac{a}{6} - \frac{1}{6}$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $a$ and $\frac{6}{6}$ .

$\frac{5 a}{6} - \frac{5}{6} = \frac{a \cdot 6}{6} + \frac{a}{6} - \frac{1}{6}$

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{5 a}{6} - \frac{5}{6} = \frac{a \cdot 6 + a}{6} - \frac{1}{6}$

Step 2.3.3

Combine the numerators over the common denominator.

$\frac{5 a}{6} - \frac{5}{6} = \frac{a \cdot 6 + a - 1}{6}$

Step 2.4

Move $6$ to the left of $a$ .

$\frac{5 a}{6} - \frac{5}{6} = \frac{6 a + a - 1}{6}$

Step 2.5

Add $6 a$ and $a$ .

$\frac{5 a}{6} - \frac{5}{6} = \frac{7 a - 1}{6}$

Step 3

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

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Step 3.1

Subtract $\frac{7 a - 1}{6}$ from both sides of the equation.

$\frac{5 a}{6} - \frac{5}{6} - \frac{7 a - 1}{6} = 0$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 a - 5 - (7 a - 1)}{6} = 0$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$\frac{5 a - 5 - (7 a) - - 1}{6} = 0$

Step 3.3.2

Multiply $7$ by $- 1$ .

$\frac{5 a - 5 - 7 a - - 1}{6} = 0$

Step 3.3.3

Multiply $- 1$ by $- 1$ .

$\frac{5 a - 5 - 7 a + 1}{6} = 0$

Step 3.4

Subtract $7 a$ from $5 a$ .

$\frac{- 2 a - 5 + 1}{6} = 0$

Step 3.5

Add $- 5$ and $1$ .

$\frac{- 2 a - 4}{6} = 0$

Step 3.6

Cancel the common factor of $- 2 a - 4$ and $6$ .

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Step 3.6.1

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

$\frac{2 (- a) - 4}{6} = 0$

Step 3.6.2

Factor $2$ out of $- 4$ .

$\frac{2 (- a) + 2 (- 2)}{6} = 0$

Step 3.6.3

Factor $2$ out of $2 (- a) + 2 (- 2)$ .

$\frac{2 (- a - 2)}{6} = 0$

Step 3.6.4

Cancel the common factors.

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Step 3.6.4.1

Factor $2$ out of $6$ .

$\frac{2 (- a - 2)}{2 (3)} = 0$

Step 3.6.4.2

Cancel the common factor.

$\frac{2 (- a - 2)}{2 \cdot 3} = 0$

Step 3.6.4.3

Rewrite the expression.

$\frac{- a - 2}{3} = 0$

Step 3.7

Factor $- 1$ out of $- a$ .

$\frac{- (a) - 2}{3} = 0$

Step 3.8

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (a) - 1 (2)}{3} = 0$

Step 3.9

Factor $- 1$ out of $- (a) - 1 (2)$ .

$\frac{- (a + 2)}{3} = 0$

Step 3.10

Rewrite $- (a + 2)$ as $- 1 (a + 2)$ .

$\frac{- 1 (a + 2)}{3} = 0$

Step 3.11

Move the negative in front of the fraction.

$- \frac{a + 2}{3} = 0$

Step 4

Set the numerator equal to zero.

$a + 2 = 0$

Step 5

Subtract $2$ from both sides of the equation.

$a = - 2$