Basic Math Examples

$2 - (- 2 (a + 1) - \frac{a - 3}{2}) = \frac{2}{3} a - \frac{5 a - 3}{12} + 3 a$

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.

$\frac{2}{3} a - \frac{5 a - 3}{12} + 3 a = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2

Simplify $\frac{2}{3} a - \frac{5 a - 3}{12} + 3 a$ .

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Step 2.1

Combine $\frac{2}{3}$ and $a$ .

$\frac{2 a}{3} - \frac{5 a - 3}{12} + 3 a = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{2 a}{3}$ by $\frac{4}{4}$ .

$\frac{2 a}{3} \cdot \frac{4}{4} - \frac{5 a - 3}{12} + 3 a = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.2.2

Multiply $\frac{2 a}{3}$ by $\frac{4}{4}$ .

$\frac{2 a \cdot 4}{3 \cdot 4} - \frac{5 a - 3}{12} + 3 a = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.2.3

Write $3 a$ as a fraction with denominator $1$ .

$\frac{2 a \cdot 4}{3 \cdot 4} - \frac{5 a - 3}{12} + \frac{3 a}{1} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.2.4

Multiply $\frac{3 a}{1}$ by $\frac{12}{12}$ .

$\frac{2 a \cdot 4}{3 \cdot 4} - \frac{5 a - 3}{12} + \frac{3 a}{1} \cdot \frac{12}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.2.5

Multiply $\frac{3 a}{1}$ by $\frac{12}{12}$ .

$\frac{2 a \cdot 4}{3 \cdot 4} - \frac{5 a - 3}{12} + \frac{3 a \cdot 12}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.2.6

Multiply $3$ by $4$ .

$\frac{2 a \cdot 4}{12} - \frac{5 a - 3}{12} + \frac{3 a \cdot 12}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.3

Combine the numerators over the common denominator.

$\frac{2 a \cdot 4 - (5 a - 3) + 3 a \cdot 12}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $4$ by $2$ .

$\frac{8 a - (5 a - 3) + 3 a \cdot 12}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.4.2

Apply the distributive property.

$\frac{8 a - (5 a) - - 3 + 3 a \cdot 12}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.4.3

Multiply $5$ by $- 1$ .

$\frac{8 a - 5 a - - 3 + 3 a \cdot 12}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.4.4

Multiply $- 1$ by $- 3$ .

$\frac{8 a - 5 a + 3 + 3 a \cdot 12}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.4.5

Multiply $12$ by $3$ .

$\frac{8 a - 5 a + 3 + 36 a}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.5

Simplify terms.

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Step 2.5.1

Subtract $5 a$ from $8 a$ .

$\frac{3 a + 3 + 36 a}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.5.2

Add $3 a$ and $36 a$ .

$\frac{39 a + 3}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.5.3

Cancel the common factor of $39 a + 3$ and $12$ .

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Step 2.5.3.1

Factor $3$ out of $39 a$ .

$\frac{3 (13 a) + 3}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.5.3.2

Factor $3$ out of $3$ .

$\frac{3 (13 a) + 3 (1)}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.5.3.3

Factor $3$ out of $3 (13 a) + 3 (1)$ .

$\frac{3 (13 a + 1)}{12} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.5.3.4

Cancel the common factors.

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Step 2.5.3.4.1

Factor $3$ out of $12$ .

$\frac{3 (13 a + 1)}{3 (4)} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.5.3.4.2

Cancel the common factor.

$\frac{3 (13 a + 1)}{3 \cdot 4} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 2.5.3.4.3

Rewrite the expression.

$\frac{13 a + 1}{4} = 2 - (- 2 (a + 1) - \frac{a - 3}{2})$

Step 3

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

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Step 3.1

Simplify each term.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$\frac{13 a + 1}{4} = 2 - (- 2 a - 2 \cdot 1 - \frac{a - 3}{2})$

Step 3.1.1.2

Multiply $- 2$ by $1$ .

$\frac{13 a + 1}{4} = 2 - (- 2 a - 2 - \frac{a - 3}{2})$

Step 3.1.2

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

$\frac{13 a + 1}{4} = 2 - (- 2 a \cdot \frac{2}{2} - \frac{a - 3}{2} - 2)$

Step 3.1.3

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

$\frac{13 a + 1}{4} = 2 - (\frac{- 2 a \cdot 2}{2} - \frac{a - 3}{2} - 2)$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{13 a + 1}{4} = 2 - (\frac{- 2 a \cdot 2 - (a - 3)}{2} - 2)$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $2$ by $- 2$ .

$\frac{13 a + 1}{4} = 2 - (\frac{- 4 a - (a - 3)}{2} - 2)$

Step 3.1.5.2

Apply the distributive property.

$\frac{13 a + 1}{4} = 2 - (\frac{- 4 a - a - - 3}{2} - 2)$

Step 3.1.5.3

Multiply $- 1$ by $- 3$ .

$\frac{13 a + 1}{4} = 2 - (\frac{- 4 a - a + 3}{2} - 2)$

Step 3.1.5.4

Subtract $a$ from $- 4 a$ .

$\frac{13 a + 1}{4} = 2 - (\frac{- 5 a + 3}{2} - 2)$

Step 3.1.6

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

$\frac{13 a + 1}{4} = 2 - (\frac{- 5 a + 3}{2} - 2 \cdot \frac{2}{2})$

Step 3.1.7

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

$\frac{13 a + 1}{4} = 2 - (\frac{- 5 a + 3}{2} + \frac{- 2 \cdot 2}{2})$

Step 3.1.8

Combine the numerators over the common denominator.

$\frac{13 a + 1}{4} = 2 - \frac{- 5 a + 3 - 2 \cdot 2}{2}$

Step 3.1.9

Simplify the numerator.

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Step 3.1.9.1

Multiply $- 2$ by $2$ .

$\frac{13 a + 1}{4} = 2 - \frac{- 5 a + 3 - 4}{2}$

Step 3.1.9.2

Subtract $4$ from $3$ .

$\frac{13 a + 1}{4} = 2 - \frac{- 5 a - 1}{2}$

Step 3.2

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

$\frac{13 a + 1}{4} = 2 \cdot \frac{2}{2} - \frac{- 5 a - 1}{2}$

Step 3.3

Simplify terms.

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Step 3.3.1

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

$\frac{13 a + 1}{4} = \frac{2 \cdot 2}{2} - \frac{- 5 a - 1}{2}$

Step 3.3.2

Combine the numerators over the common denominator.

$\frac{13 a + 1}{4} = \frac{2 \cdot 2 - (- 5 a - 1)}{2}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $2$ by $2$ .

$\frac{13 a + 1}{4} = \frac{4 - (- 5 a - 1)}{2}$

Step 3.4.2

Apply the distributive property.

$\frac{13 a + 1}{4} = \frac{4 - (- 5 a) - - 1}{2}$

Step 3.4.3

Multiply $- 5$ by $- 1$ .

$\frac{13 a + 1}{4} = \frac{4 + 5 a - - 1}{2}$

Step 3.4.4

Multiply $- 1$ by $- 1$ .

$\frac{13 a + 1}{4} = \frac{4 + 5 a + 1}{2}$

Step 3.4.5

Add $4$ and $1$ .

$\frac{13 a + 1}{4} = \frac{5 a + 5}{2}$

Step 3.4.6

Factor $5$ out of $5 a + 5$ .

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Step 3.4.6.1

Factor $5$ out of $5 a$ .

$\frac{13 a + 1}{4} = \frac{5 (a) + 5}{2}$

Step 3.4.6.2

Factor $5$ out of $5$ .

$\frac{13 a + 1}{4} = \frac{5 (a) + 5 (1)}{2}$

Step 3.4.6.3

Factor $5$ out of $5 (a) + 5 (1)$ .

$\frac{13 a + 1}{4} = \frac{5 (a + 1)}{2}$

Step 4

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

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Step 4.1

Subtract $\frac{5 (a + 1)}{2}$ from both sides of the equation.

$\frac{13 a + 1}{4} - \frac{5 (a + 1)}{2} = 0$

Step 4.2

To write $- \frac{5 (a + 1)}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{13 a + 1}{4} - \frac{5 (a + 1)}{2} \cdot \frac{2}{2} = 0$

Step 4.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{5 (a + 1)}{2}$ by $\frac{2}{2}$ .

$\frac{13 a + 1}{4} - \frac{5 (a + 1) \cdot 2}{2 \cdot 2} = 0$

Step 4.3.2

Multiply $2$ by $2$ .

$\frac{13 a + 1}{4} - \frac{5 (a + 1) \cdot 2}{4} = 0$

Step 4.4

Combine the numerators over the common denominator.

$\frac{13 a + 1 - 5 (a + 1) \cdot 2}{4} = 0$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Apply the distributive property.

$\frac{13 a + 1 + (- 5 a - 5 \cdot 1) \cdot 2}{4} = 0$

Step 4.5.2

Multiply $- 5$ by $1$ .

$\frac{13 a + 1 + (- 5 a - 5) \cdot 2}{4} = 0$

Step 4.5.3

Apply the distributive property.

$\frac{13 a + 1 - 5 a \cdot 2 - 5 \cdot 2}{4} = 0$

Step 4.5.4

Multiply $2$ by $- 5$ .

$\frac{13 a + 1 - 10 a - 5 \cdot 2}{4} = 0$

Step 4.5.5

Multiply $- 5$ by $2$ .

$\frac{13 a + 1 - 10 a - 10}{4} = 0$

Step 4.5.6

Subtract $10 a$ from $13 a$ .

$\frac{3 a + 1 - 10}{4} = 0$

Step 4.5.7

Subtract $10$ from $1$ .

$\frac{3 a - 9}{4} = 0$

Step 4.5.8

Factor $3$ out of $3 a - 9$ .

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Step 4.5.8.1

Factor $3$ out of $3 a$ .

$\frac{3 (a) - 9}{4} = 0$

Step 4.5.8.2

Factor $3$ out of $- 9$ .

$\frac{3 a + 3 \cdot - 3}{4} = 0$

Step 4.5.8.3

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

$\frac{3 (a - 3)}{4} = 0$

Step 5

Set the numerator equal to zero.

$3 (a - 3) = 0$

Step 6

Solve the equation for $a$ .

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Step 6.1

Divide each term in $3 (a - 3) = 0$ by $3$ and simplify.

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Step 6.1.1

Divide each term in $3 (a - 3) = 0$ by $3$ .

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

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $3$ .

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Step 6.1.2.1.1

Cancel the common factor.

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

Step 6.1.2.1.2

Divide $a - 3$ by $1$ .

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

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $0$ by $3$ .

$a - 3 = 0$

Step 6.2

Add $3$ to both sides of the equation.

$a = 3$