Algebra Examples

$\frac{1}{2} a + \frac{1}{3} b + 2 a - 3 b - \frac{3}{4} a - \frac{1}{6} b + \frac{3}{4} - \frac{1}{2}$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{a}{2} + \frac{1}{3} b + 2 a - 3 b - \frac{3}{4} a - \frac{1}{6} b + \frac{3}{4} - \frac{1}{2}$

Step 1.2

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

$\frac{a}{2} + \frac{b}{3} + 2 a - 3 b - \frac{3}{4} a - \frac{1}{6} b + \frac{3}{4} - \frac{1}{2}$

Step 1.3

Combine $a$ and $\frac{3}{4}$ .

$\frac{a}{2} + \frac{b}{3} + 2 a - 3 b - \frac{a \cdot 3}{4} - \frac{1}{6} b + \frac{3}{4} - \frac{1}{2}$

Step 1.4

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

$\frac{a}{2} + \frac{b}{3} + 2 a - 3 b - \frac{3 \cdot a}{4} - \frac{1}{6} b + \frac{3}{4} - \frac{1}{2}$

Step 1.5

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

$\frac{a}{2} + \frac{b}{3} + 2 a - 3 b - \frac{3 a}{4} - \frac{b}{6} + \frac{3}{4} - \frac{1}{2}$

Step 2

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

$\frac{b}{3} + \frac{a}{2} + 2 a \cdot \frac{2}{2} - 3 b - \frac{3 a}{4} - \frac{b}{6} + \frac{3}{4} - \frac{1}{2}$

Step 3

Simplify terms.

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Step 3.1

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

$\frac{b}{3} + \frac{a}{2} + \frac{2 a \cdot 2}{2} - 3 b - \frac{3 a}{4} - \frac{b}{6} + \frac{3}{4} - \frac{1}{2}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{b}{3} + \frac{a + 2 a \cdot 2}{2} - 3 b - \frac{3 a}{4} - \frac{b}{6} + \frac{3}{4} - \frac{1}{2}$

Step 3.3

Combine the numerators over the common denominator.

$- 3 b + \frac{b}{3} + \frac{a + 2 a \cdot 2 - 1}{2} + \frac{- 3 a}{4} + \frac{3}{4} + \frac{- b}{6}$

Step 3.4

Multiply $2$ by $2$ .

$- 3 b + \frac{b}{3} + \frac{a + 4 a - 1}{2} + \frac{- 3 a}{4} + \frac{3}{4} + \frac{- b}{6}$

Step 3.5

Add $a$ and $4 a$ .

$- 3 b + \frac{b}{3} + \frac{5 a - 1}{2} + \frac{- 3 a}{4} + \frac{3}{4} + \frac{- b}{6}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- 3 b + \frac{b}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} + \frac{- b}{6}$

Step 4.2

Move the negative in front of the fraction.

$- 3 b + \frac{b}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 5

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

$- 3 b \cdot \frac{3}{3} + \frac{b}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 6

Simplify terms.

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Step 6.1

Combine $- 3 b$ and $\frac{3}{3}$ .

$\frac{- 3 b \cdot 3}{3} + \frac{b}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{- 3 b \cdot 3 + b}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 7

Simplify each term.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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Step 7.1.1.1

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

$\frac{b (- 3 \cdot 3) + b}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 7.1.1.2

Raise $b$ to the power of $1$ .

$\frac{b (- 3 \cdot 3) + b^{1}}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 7.1.1.3

Factor $b$ out of $b^{1}$ .

$\frac{b (- 3 \cdot 3) + b \cdot 1}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 7.1.1.4

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

$\frac{b (- 3 \cdot 3 + 1)}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 7.1.2

Multiply $- 3$ by $3$ .

$\frac{b (- 9 + 1)}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 7.1.3

Add $- 9$ and $1$ .

$\frac{b \cdot - 8}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 7.2

Move $- 8$ to the left of $b$ .

$\frac{- 8 \cdot b}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 7.3

Move the negative in front of the fraction.

$- \frac{8 b}{3} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 8

To write $- \frac{8 b}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{8 b}{3} \cdot \frac{2}{2} + \frac{5 a - 1}{2} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 9

To write $\frac{5 a - 1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{8 b}{3} \cdot \frac{2}{2} + \frac{5 a - 1}{2} \cdot \frac{3}{3} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 10

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

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Step 10.1

Multiply $\frac{8 b}{3}$ by $\frac{2}{2}$ .

$- \frac{8 b \cdot 2}{3 \cdot 2} + \frac{5 a - 1}{2} \cdot \frac{3}{3} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 10.2

Multiply $3$ by $2$ .

$- \frac{8 b \cdot 2}{6} + \frac{5 a - 1}{2} \cdot \frac{3}{3} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 10.3

Multiply $\frac{5 a - 1}{2}$ by $\frac{3}{3}$ .

$- \frac{8 b \cdot 2}{6} + \frac{(5 a - 1) \cdot 3}{2 \cdot 3} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 10.4

Multiply $2$ by $3$ .

$- \frac{8 b \cdot 2}{6} + \frac{(5 a - 1) \cdot 3}{6} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 11

Simplify terms.

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Step 11.1

Combine the numerators over the common denominator.

$\frac{- 8 b \cdot 2 + (5 a - 1) \cdot 3}{6} - \frac{3 a}{4} + \frac{3}{4} - \frac{b}{6}$

Step 11.2

Combine the numerators over the common denominator.

$\frac{- 8 b \cdot 2 + (5 a - 1) \cdot 3 - b}{6} + \frac{- 3 a}{4} + \frac{3}{4}$

Step 12

Simplify each term.

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Step 12.1

Multiply $2$ by $- 8$ .

$\frac{- 16 b + (5 a - 1) \cdot 3 - b}{6} + \frac{- 3 a}{4} + \frac{3}{4}$

Step 12.2

Apply the distributive property.

$\frac{- 16 b + 5 a \cdot 3 - 1 \cdot 3 - b}{6} + \frac{- 3 a}{4} + \frac{3}{4}$

Step 12.3

Multiply $3$ by $5$ .

$\frac{- 16 b + 15 a - 1 \cdot 3 - b}{6} + \frac{- 3 a}{4} + \frac{3}{4}$

Step 12.4

Multiply $- 1$ by $3$ .

$\frac{- 16 b + 15 a - 3 - b}{6} + \frac{- 3 a}{4} + \frac{3}{4}$

Step 13

Simplify by adding terms.

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Step 13.1

Subtract $b$ from $- 16 b$ .

$\frac{- 17 b + 15 a - 3}{6} + \frac{- 3 a}{4} + \frac{3}{4}$

Step 13.2

Simplify the expression.

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Step 13.2.1

Reorder $- 17 b$ and $15 a$ .

$\frac{15 a - 17 b - 3}{6} + \frac{- 3 a}{4} + \frac{3}{4}$

Step 13.2.2

Move the negative in front of the fraction.

$\frac{15 a - 17 b - 3}{6} - \frac{3 a}{4} + \frac{3}{4}$

Step 14

To write $\frac{15 a - 17 b - 3}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{15 a - 17 b - 3}{6} \cdot \frac{2}{2} - \frac{3 a}{4} + \frac{3}{4}$

Step 15

To write $- \frac{3 a}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{15 a - 17 b - 3}{6} \cdot \frac{2}{2} - \frac{3 a}{4} \cdot \frac{3}{3} + \frac{3}{4}$

Step 16

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.1

Multiply $\frac{15 a - 17 b - 3}{6}$ by $\frac{2}{2}$ .

$\frac{(15 a - 17 b - 3) \cdot 2}{6 \cdot 2} - \frac{3 a}{4} \cdot \frac{3}{3} + \frac{3}{4}$

Step 16.2

Multiply $6$ by $2$ .

$\frac{(15 a - 17 b - 3) \cdot 2}{12} - \frac{3 a}{4} \cdot \frac{3}{3} + \frac{3}{4}$

Step 16.3

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

$\frac{(15 a - 17 b - 3) \cdot 2}{12} - \frac{3 a \cdot 3}{4 \cdot 3} + \frac{3}{4}$

Step 16.4

Multiply $4$ by $3$ .

$\frac{(15 a - 17 b - 3) \cdot 2}{12} - \frac{3 a \cdot 3}{12} + \frac{3}{4}$

Step 17

Combine the numerators over the common denominator.

$\frac{(15 a - 17 b - 3) \cdot 2 - 3 a \cdot 3}{12} + \frac{3}{4}$

Step 18

Simplify the numerator.

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Step 18.1

Apply the distributive property.

$\frac{15 a \cdot 2 - 17 b \cdot 2 - 3 \cdot 2 - 3 a \cdot 3}{12} + \frac{3}{4}$

Step 18.2

Simplify.

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Step 18.2.1

Multiply $2$ by $15$ .

$\frac{30 a - 17 b \cdot 2 - 3 \cdot 2 - 3 a \cdot 3}{12} + \frac{3}{4}$

Step 18.2.2

Multiply $2$ by $- 17$ .

$\frac{30 a - 34 b - 3 \cdot 2 - 3 a \cdot 3}{12} + \frac{3}{4}$

Step 18.2.3

Multiply $- 3$ by $2$ .

$\frac{30 a - 34 b - 6 - 3 a \cdot 3}{12} + \frac{3}{4}$

Step 18.3

Multiply $3$ by $- 3$ .

$\frac{30 a - 34 b - 6 - 9 a}{12} + \frac{3}{4}$

Step 18.4

Subtract $9 a$ from $30 a$ .

$\frac{21 a - 34 b - 6}{12} + \frac{3}{4}$

Step 19

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{21 a - 34 b - 6}{12} + \frac{3}{4} \cdot \frac{3}{3}$

Step 20

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 20.1

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

$\frac{21 a - 34 b - 6}{12} + \frac{3 \cdot 3}{4 \cdot 3}$

Step 20.2

Multiply $4$ by $3$ .

$\frac{21 a - 34 b - 6}{12} + \frac{3 \cdot 3}{12}$

Step 21

Combine the numerators over the common denominator.

$\frac{21 a - 34 b - 6 + 3 \cdot 3}{12}$

Step 22

Simplify the numerator.

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Step 22.1

Multiply $3$ by $3$ .

$\frac{21 a - 34 b - 6 + 9}{12}$

Step 22.2

Add $- 6$ and $9$ .

$\frac{21 a - 34 b + 3}{12}$