Basic Math Examples

$\frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36} = 3 v - 20 - \frac{8 - v}{12} - \frac{20 - 3 v}{18}$

Step 1

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

$3 v - 20 - \frac{8 - v}{12} - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2

Simplify $3 v - 20 - \frac{8 - v}{12} - \frac{20 - 3 v}{18}$ .

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Step 2.1

To write $3 v$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$3 v \cdot \frac{12}{12} - \frac{8 - v}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $3 v$ and $\frac{12}{12}$ .

$\frac{3 v \cdot 12}{12} - \frac{8 - v}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{3 v \cdot 12 - (8 - v)}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Multiply $12$ by $3$ .

$\frac{36 v - (8 - v)}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.3.2

Apply the distributive property.

$\frac{36 v - 1 \cdot 8 - - v}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.3.3

Multiply $- 1$ by $8$ .

$\frac{36 v - 8 - - v}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.3.4

Multiply $- - v$ .

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

$\frac{36 v - 8 + 1 v}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.3.4.2

Multiply $v$ by $1$ .

$\frac{36 v - 8 + v}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.3.5

Add $36 v$ and $v$ .

$\frac{37 v - 8}{12} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4

Find the common denominator.

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Step 2.4.1

Multiply $\frac{37 v - 8}{12}$ by $\frac{3}{3}$ .

$\frac{37 v - 8}{12} \cdot \frac{3}{3} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.2

Multiply $\frac{37 v - 8}{12}$ by $\frac{3}{3}$ .

$\frac{(37 v - 8) \cdot 3}{12 \cdot 3} - 20 - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.3

Write $- 20$ as a fraction with denominator $1$ .

$\frac{(37 v - 8) \cdot 3}{12 \cdot 3} + \frac{- 20}{1} - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.4

Multiply $\frac{- 20}{1}$ by $\frac{36}{36}$ .

$\frac{(37 v - 8) \cdot 3}{12 \cdot 3} + \frac{- 20}{1} \cdot \frac{36}{36} - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.5

Multiply $\frac{- 20}{1}$ by $\frac{36}{36}$ .

$\frac{(37 v - 8) \cdot 3}{12 \cdot 3} + \frac{- 20 \cdot 36}{36} - \frac{20 - 3 v}{18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.6

Multiply $\frac{20 - 3 v}{18}$ by $\frac{2}{2}$ .

$\frac{(37 v - 8) \cdot 3}{12 \cdot 3} + \frac{- 20 \cdot 36}{36} - (\frac{20 - 3 v}{18} \cdot \frac{2}{2}) = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.7

Multiply $\frac{20 - 3 v}{18}$ by $\frac{2}{2}$ .

$\frac{(37 v - 8) \cdot 3}{12 \cdot 3} + \frac{- 20 \cdot 36}{36} - \frac{(20 - 3 v) \cdot 2}{18 \cdot 2} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.8

Reorder the factors of $12 \cdot 3$ .

$\frac{(37 v - 8) \cdot 3}{3 \cdot 12} + \frac{- 20 \cdot 36}{36} - \frac{(20 - 3 v) \cdot 2}{18 \cdot 2} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.9

Multiply $3$ by $12$ .

$\frac{(37 v - 8) \cdot 3}{36} + \frac{- 20 \cdot 36}{36} - \frac{(20 - 3 v) \cdot 2}{18 \cdot 2} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.10

Reorder the factors of $18 \cdot 2$ .

$\frac{(37 v - 8) \cdot 3}{36} + \frac{- 20 \cdot 36}{36} - \frac{(20 - 3 v) \cdot 2}{2 \cdot 18} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.4.11

Multiply $2$ by $18$ .

$\frac{(37 v - 8) \cdot 3}{36} + \frac{- 20 \cdot 36}{36} - \frac{(20 - 3 v) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{(37 v - 8) \cdot 3 - 20 \cdot 36 - (20 - 3 v) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6

Simplify each term.

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Step 2.6.1

Apply the distributive property.

$\frac{37 v \cdot 3 - 8 \cdot 3 - 20 \cdot 36 - (20 - 3 v) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.2

Multiply $3$ by $37$ .

$\frac{111 v - 8 \cdot 3 - 20 \cdot 36 - (20 - 3 v) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.3

Multiply $- 8$ by $3$ .

$\frac{111 v - 24 - 20 \cdot 36 - (20 - 3 v) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.4

Multiply $- 20$ by $36$ .

$\frac{111 v - 24 - 720 - (20 - 3 v) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.5

Apply the distributive property.

$\frac{111 v - 24 - 720 + (- 1 \cdot 20 - (- 3 v)) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.6

Multiply $- 1$ by $20$ .

$\frac{111 v - 24 - 720 + (- 20 - (- 3 v)) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.7

Multiply $- 3$ by $- 1$ .

$\frac{111 v - 24 - 720 + (- 20 + 3 v) \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.8

Apply the distributive property.

$\frac{111 v - 24 - 720 - 20 \cdot 2 + 3 v \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.9

Multiply $- 20$ by $2$ .

$\frac{111 v - 24 - 720 - 40 + 3 v \cdot 2}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.6.10

Multiply $2$ by $3$ .

$\frac{111 v - 24 - 720 - 40 + 6 v}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.7

Simplify by adding terms.

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Step 2.7.1

Add $111 v$ and $6 v$ .

$\frac{117 v - 24 - 720 - 40}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.7.2

Simplify by subtracting numbers.

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Step 2.7.2.1

Subtract $720$ from $- 24$ .

$\frac{117 v - 744 - 40}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 2.7.2.2

Subtract $40$ from $- 744$ .

$\frac{117 v - 784}{36} = \frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$

Step 3

Simplify $\frac{5 (v + 2)}{12} + \frac{4}{9} - \frac{22 - v}{36}$ .

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Step 3.1

Find the common denominator.

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Step 3.1.1

Multiply $\frac{5 (v + 2)}{12}$ by $\frac{3}{3}$ .

$\frac{117 v - 784}{36} = \frac{5 (v + 2)}{12} \cdot \frac{3}{3} + \frac{4}{9} - \frac{22 - v}{36}$

Step 3.1.2

Multiply $\frac{5 (v + 2)}{12}$ by $\frac{3}{3}$ .

$\frac{117 v - 784}{36} = \frac{5 (v + 2) \cdot 3}{12 \cdot 3} + \frac{4}{9} - \frac{22 - v}{36}$

Step 3.1.3

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

$\frac{117 v - 784}{36} = \frac{5 (v + 2) \cdot 3}{12 \cdot 3} + \frac{4}{9} \cdot \frac{4}{4} - \frac{22 - v}{36}$

Step 3.1.4

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

$\frac{117 v - 784}{36} = \frac{5 (v + 2) \cdot 3}{12 \cdot 3} + \frac{4 \cdot 4}{9 \cdot 4} - \frac{22 - v}{36}$

Step 3.1.5

Reorder the factors of $12 \cdot 3$ .

$\frac{117 v - 784}{36} = \frac{5 (v + 2) \cdot 3}{3 \cdot 12} + \frac{4 \cdot 4}{9 \cdot 4} - \frac{22 - v}{36}$

Step 3.1.6

Multiply $3$ by $12$ .

$\frac{117 v - 784}{36} = \frac{5 (v + 2) \cdot 3}{36} + \frac{4 \cdot 4}{9 \cdot 4} - \frac{22 - v}{36}$

Step 3.1.7

Reorder the factors of $9 \cdot 4$ .

$\frac{117 v - 784}{36} = \frac{5 (v + 2) \cdot 3}{36} + \frac{4 \cdot 4}{4 \cdot 9} - \frac{22 - v}{36}$

Step 3.1.8

Multiply $4$ by $9$ .

$\frac{117 v - 784}{36} = \frac{5 (v + 2) \cdot 3}{36} + \frac{4 \cdot 4}{36} - \frac{22 - v}{36}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{117 v - 784}{36} = \frac{5 (v + 2) \cdot 3 + 4 \cdot 4 - (22 - v)}{36}$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$\frac{117 v - 784}{36} = \frac{(5 v + 5 \cdot 2) \cdot 3 + 4 \cdot 4 - (22 - v)}{36}$

Step 3.3.2

Multiply $5$ by $2$ .

$\frac{117 v - 784}{36} = \frac{(5 v + 10) \cdot 3 + 4 \cdot 4 - (22 - v)}{36}$

Step 3.3.3

Apply the distributive property.

$\frac{117 v - 784}{36} = \frac{5 v \cdot 3 + 10 \cdot 3 + 4 \cdot 4 - (22 - v)}{36}$

Step 3.3.4

Multiply $3$ by $5$ .

$\frac{117 v - 784}{36} = \frac{15 v + 10 \cdot 3 + 4 \cdot 4 - (22 - v)}{36}$

Step 3.3.5

Multiply $10$ by $3$ .

$\frac{117 v - 784}{36} = \frac{15 v + 30 + 4 \cdot 4 - (22 - v)}{36}$

Step 3.3.6

Multiply $4$ by $4$ .

$\frac{117 v - 784}{36} = \frac{15 v + 30 + 16 - (22 - v)}{36}$

Step 3.3.7

Apply the distributive property.

$\frac{117 v - 784}{36} = \frac{15 v + 30 + 16 - 1 \cdot 22 - - v}{36}$

Step 3.3.8

Multiply $- 1$ by $22$ .

$\frac{117 v - 784}{36} = \frac{15 v + 30 + 16 - 22 - - v}{36}$

Step 3.3.9

Multiply $- - v$ .

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Step 3.3.9.1

Multiply $- 1$ by $- 1$ .

$\frac{117 v - 784}{36} = \frac{15 v + 30 + 16 - 22 + 1 v}{36}$

Step 3.3.9.2

Multiply $v$ by $1$ .

$\frac{117 v - 784}{36} = \frac{15 v + 30 + 16 - 22 + v}{36}$

Step 3.4

Simplify terms.

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Step 3.4.1

Add $15 v$ and $v$ .

$\frac{117 v - 784}{36} = \frac{16 v + 30 + 16 - 22}{36}$

Step 3.4.2

Simplify by adding and subtracting.

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Step 3.4.2.1

Add $30$ and $16$ .

$\frac{117 v - 784}{36} = \frac{16 v + 46 - 22}{36}$

Step 3.4.2.2

Subtract $22$ from $46$ .

$\frac{117 v - 784}{36} = \frac{16 v + 24}{36}$

Step 3.4.3

Cancel the common factor of $16 v + 24$ and $36$ .

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Step 3.4.3.1

Factor $4$ out of $16 v$ .

$\frac{117 v - 784}{36} = \frac{4 (4 v) + 24}{36}$

Step 3.4.3.2

Factor $4$ out of $24$ .

$\frac{117 v - 784}{36} = \frac{4 (4 v) + 4 (6)}{36}$

Step 3.4.3.3

Factor $4$ out of $4 (4 v) + 4 (6)$ .

$\frac{117 v - 784}{36} = \frac{4 (4 v + 6)}{36}$

Step 3.4.3.4

Cancel the common factors.

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Step 3.4.3.4.1

Factor $4$ out of $36$ .

$\frac{117 v - 784}{36} = \frac{4 (4 v + 6)}{4 (9)}$

Step 3.4.3.4.2

Cancel the common factor.

$\frac{117 v - 784}{36} = \frac{4 (4 v + 6)}{4 \cdot 9}$

Step 3.4.3.4.3

Rewrite the expression.

$\frac{117 v - 784}{36} = \frac{4 v + 6}{9}$

Step 3.4.4

Factor $2$ out of $4 v + 6$ .

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Step 3.4.4.1

Factor $2$ out of $4 v$ .

$\frac{117 v - 784}{36} = \frac{2 (2 v) + 6}{9}$

Step 3.4.4.2

Factor $2$ out of $6$ .

$\frac{117 v - 784}{36} = \frac{2 (2 v) + 2 (3)}{9}$

Step 3.4.4.3

Factor $2$ out of $2 (2 v) + 2 (3)$ .

$\frac{117 v - 784}{36} = \frac{2 (2 v + 3)}{9}$

Step 4

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

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Step 4.1

Subtract $\frac{2 (2 v + 3)}{9}$ from both sides of the equation.

$\frac{117 v - 784}{36} - \frac{2 (2 v + 3)}{9} = 0$

Step 4.2

To write $- \frac{2 (2 v + 3)}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{117 v - 784}{36} - \frac{2 (2 v + 3)}{9} \cdot \frac{4}{4} = 0$

Step 4.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{2 (2 v + 3)}{9}$ by $\frac{4}{4}$ .

$\frac{117 v - 784}{36} - \frac{2 (2 v + 3) \cdot 4}{9 \cdot 4} = 0$

Step 4.3.2

Multiply $9$ by $4$ .

$\frac{117 v - 784}{36} - \frac{2 (2 v + 3) \cdot 4}{36} = 0$

Step 4.4

Combine the numerators over the common denominator.

$\frac{117 v - 784 - 2 (2 v + 3) \cdot 4}{36} = 0$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Apply the distributive property.

$\frac{117 v - 784 + (- 2 (2 v) - 2 \cdot 3) \cdot 4}{36} = 0$

Step 4.5.2

Multiply $2$ by $- 2$ .

$\frac{117 v - 784 + (- 4 v - 2 \cdot 3) \cdot 4}{36} = 0$

Step 4.5.3

Multiply $- 2$ by $3$ .

$\frac{117 v - 784 + (- 4 v - 6) \cdot 4}{36} = 0$

Step 4.5.4

Apply the distributive property.

$\frac{117 v - 784 - 4 v \cdot 4 - 6 \cdot 4}{36} = 0$

Step 4.5.5

Multiply $4$ by $- 4$ .

$\frac{117 v - 784 - 16 v - 6 \cdot 4}{36} = 0$

Step 4.5.6

Multiply $- 6$ by $4$ .

$\frac{117 v - 784 - 16 v - 24}{36} = 0$

Step 4.5.7

Subtract $16 v$ from $117 v$ .

$\frac{101 v - 784 - 24}{36} = 0$

Step 4.5.8

Subtract $24$ from $- 784$ .

$\frac{101 v - 808}{36} = 0$

Step 4.5.9

Factor $101$ out of $101 v - 808$ .

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Step 4.5.9.1

Factor $101$ out of $101 v$ .

$\frac{101 (v) - 808}{36} = 0$

Step 4.5.9.2

Factor $101$ out of $- 808$ .

$\frac{101 v + 101 \cdot - 8}{36} = 0$

Step 4.5.9.3

Factor $101$ out of $101 v + 101 \cdot - 8$ .

$\frac{101 (v - 8)}{36} = 0$

Step 5

Set the numerator equal to zero.

$101 (v - 8) = 0$

Step 6

Solve the equation for $v$ .

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Step 6.1

Divide each term in $101 (v - 8) = 0$ by $101$ and simplify.

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Step 6.1.1

Divide each term in $101 (v - 8) = 0$ by $101$ .

$\frac{101 (v - 8)}{101} = \frac{0}{101}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $101$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{101 (v - 8)}{101} = \frac{0}{101}$

Step 6.1.2.1.2

Divide $v - 8$ by $1$ .

$v - 8 = \frac{0}{101}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $0$ by $101$ .

$v - 8 = 0$

Step 6.2

Add $8$ to both sides of the equation.

$v = 8$