Finite Math Examples

$(1 - 2 i) x + (1 + 2 i) y = 1 + i$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$1 x - 2 i x + (1 + 2 i) y = 1 + i$

Step 1.2

Multiply $x$ by $1$ .

$x - 2 i x + (1 + 2 i) y = 1 + i$

Step 1.3

Apply the distributive property.

$x - 2 i x + 1 y + 2 i y = 1 + i$

Step 1.4

Multiply $y$ by $1$ .

$x - 2 i x + y + 2 i y = 1 + i$

Step 2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.1

Subtract $x$ from both sides of the equation.

$- 2 i x + y + 2 i y = 1 + i - x$

Step 2.2

Add $2 i x$ to both sides of the equation.

$y + 2 i y = 1 + i - x + 2 i x$

Step 3

Factor $y$ out of $y + 2 i y$ .

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Step 3.1

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

$y \cdot 1 + 2 i y = 1 + i - x + 2 i x$

Step 3.2

Factor $y$ out of $2 i y$ .

$y \cdot 1 + y (2 i) = 1 + i - x + 2 i x$

Step 3.3

Factor $y$ out of $y \cdot 1 + y (2 i)$ .

$y (1 + 2 i) = 1 + i - x + 2 i x$

Step 4

Divide each term in $y (1 + 2 i) = 1 + i - x + 2 i x$ by $1 + 2 i$ and simplify.

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Step 4.1

Divide each term in $y (1 + 2 i) = 1 + i - x + 2 i x$ by $1 + 2 i$ .

$\frac{y (1 + 2 i)}{1 + 2 i} = \frac{1}{1 + 2 i} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $1 + 2 i$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{y (1 + 2 i)}{1 + 2 i} = \frac{1}{1 + 2 i} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{1 + 2 i} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply the numerator and denominator of $\frac{1}{1 + 2 i}$ by the conjugate of $1 + 2 i$ to make the denominator real.

$y = \frac{1}{1 + 2 i} \cdot \frac{1 - 2 i}{1 - 2 i} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2

Multiply.

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Step 4.3.1.2.1

Combine.

$y = \frac{1 (1 - 2 i)}{(1 + 2 i) (1 - 2 i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.2

Multiply $1 - 2 i$ by $1$ .

$y = \frac{1 - 2 i}{(1 + 2 i) (1 - 2 i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3

Simplify the denominator.

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Step 4.3.1.2.3.1

Expand $(1 + 2 i) (1 - 2 i)$ using the FOIL Method.

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Step 4.3.1.2.3.1.1

Apply the distributive property.

$y = \frac{1 - 2 i}{1 (1 - 2 i) + 2 i (1 - 2 i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.1.2

Apply the distributive property.

$y = \frac{1 - 2 i}{1 \cdot 1 + 1 (- 2 i) + 2 i (1 - 2 i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.1.3

Apply the distributive property.

$y = \frac{1 - 2 i}{1 \cdot 1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2

Simplify.

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Step 4.3.1.2.3.2.1

Multiply $1$ by $1$ .

$y = \frac{1 - 2 i}{1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.2

Multiply $- 2$ by $1$ .

$y = \frac{1 - 2 i}{1 - 2 i + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.3

Multiply $2$ by $1$ .

$y = \frac{1 - 2 i}{1 - 2 i + 2 i + 2 i (- 2 i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.4

Multiply $- 2$ by $2$ .

$y = \frac{1 - 2 i}{1 - 2 i + 2 i - 4 i i} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.5

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

$y = \frac{1 - 2 i}{1 - 2 i + 2 i - 4 (i^{1} i)} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.6

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

$y = \frac{1 - 2 i}{1 - 2 i + 2 i - 4 (i^{1} i^{1})} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 - 2 i}{1 - 2 i + 2 i - 4 i^{1 + 1}} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.8

Add $1$ and $1$ .

$y = \frac{1 - 2 i}{1 - 2 i + 2 i - 4 i^{2}} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.9

Add $- 2 i$ and $2 i$ .

$y = \frac{1 - 2 i}{1 + 0 - 4 i^{2}} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.2.10

Add $1$ and $0$ .

$y = \frac{1 - 2 i}{1 - 4 i^{2}} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.3

Simplify each term.

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Step 4.3.1.2.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1 - 2 i}{1 - 4 \cdot - 1} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{1 - 2 i}{1 + 4} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.2.3.4

Add $1$ and $4$ .

$y = \frac{1 - 2 i}{5} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.3

Split the fraction $\frac{1 - 2 i}{5}$ into two fractions.

$y = \frac{1}{5} + \frac{- 2 i}{5} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.4

Move the negative in front of the fraction.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i}{1 + 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.5

Multiply the numerator and denominator of $\frac{i}{1 + 2 i}$ by the conjugate of $1 + 2 i$ to make the denominator real.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i}{1 + 2 i} \cdot \frac{1 - 2 i}{1 - 2 i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6

Multiply.

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Step 4.3.1.6.1

Combine.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i (1 - 2 i)}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2

Simplify the numerator.

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Step 4.3.1.6.2.1

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i \cdot 1 + i (- 2 i)}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2.2

Multiply $i$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i + i (- 2 i)}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2.3

Multiply $i (- 2 i)$ .

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Step 4.3.1.6.2.3.1

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i - 2 (i^{1} i)}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2.3.2

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i - 2 (i^{1} i^{1})}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i - 2 i^{1 + 1}}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2.3.4

Add $1$ and $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i - 2 i^{2}}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2.4

Simplify each term.

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Step 4.3.1.6.2.4.1

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i - 2 \cdot - 1}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2.4.2

Multiply $- 2$ by $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{i + 2}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.2.5

Reorder $i$ and $2$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{(1 + 2 i) (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3

Simplify the denominator.

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Step 4.3.1.6.3.1

Expand $(1 + 2 i) (1 - 2 i)$ using the FOIL Method.

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Step 4.3.1.6.3.1.1

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 (1 - 2 i) + 2 i (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.1.2

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 \cdot 1 + 1 (- 2 i) + 2 i (1 - 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.1.3

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 \cdot 1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2

Simplify.

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Step 4.3.1.6.3.2.1

Multiply $1$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.2

Multiply $- 2$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 2 i + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.3

Multiply $2$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 2 i + 2 i + 2 i (- 2 i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.4

Multiply $- 2$ by $2$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 2 i + 2 i - 4 i i} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.5

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 2 i + 2 i - 4 (i^{1} i)} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.6

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 2 i + 2 i - 4 (i^{1} i^{1})} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 2 i + 2 i - 4 i^{1 + 1}} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.8

Add $1$ and $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 2 i + 2 i - 4 i^{2}} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.9

Add $- 2 i$ and $2 i$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 + 0 - 4 i^{2}} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.2.10

Add $1$ and $0$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 4 i^{2}} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.3

Simplify each term.

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Step 4.3.1.6.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 - 4 \cdot - 1} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{1 + 4} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.6.3.4

Add $1$ and $4$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2 + i}{5} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.7

Split the fraction $\frac{2 + i}{5}$ into two fractions.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x}{1 + 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.8

Multiply the numerator and denominator of $\frac{- x}{1 + 2 i}$ by the conjugate of $1 + 2 i$ to make the denominator real.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x}{1 + 2 i} \cdot \frac{1 - 2 i}{1 - 2 i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9

Multiply.

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Step 4.3.1.9.1

Combine.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x (1 - 2 i)}{(1 + 2 i) (1 - 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.2

Simplify the numerator.

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Step 4.3.1.9.2.1

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x \cdot 1 - x (- 2 i)}{(1 + 2 i) (1 - 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.2.2

Multiply $- 1$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x - x (- 2 i)}{(1 + 2 i) (1 - 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.2.3

Multiply $- 2$ by $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{(1 + 2 i) (1 - 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3

Simplify the denominator.

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Step 4.3.1.9.3.1

Expand $(1 + 2 i) (1 - 2 i)$ using the FOIL Method.

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Step 4.3.1.9.3.1.1

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 (1 - 2 i) + 2 i (1 - 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.1.2

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 \cdot 1 + 1 (- 2 i) + 2 i (1 - 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.1.3

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 \cdot 1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2

Simplify.

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Step 4.3.1.9.3.2.1

Multiply $1$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.2

Multiply $- 2$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 2 i + 2 i \cdot 1 + 2 i (- 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.3

Multiply $2$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 2 i + 2 i + 2 i (- 2 i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.4

Multiply $- 2$ by $2$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 2 i + 2 i - 4 i i} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.5

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 2 i + 2 i - 4 (i^{1} i)} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.6

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 2 i + 2 i - 4 (i^{1} i^{1})} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 2 i + 2 i - 4 i^{1 + 1}} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.8

Add $1$ and $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 2 i + 2 i - 4 i^{2}} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.9

Add $- 2 i$ and $2 i$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 + 0 - 4 i^{2}} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.2.10

Add $1$ and $0$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 4 i^{2}} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.3

Simplify each term.

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Step 4.3.1.9.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 - 4 \cdot - 1} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{1 + 4} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.9.3.4

Add $1$ and $4$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{- x + 2 x i}{5} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.10

Factor $x$ out of $- x + 2 x i$ .

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Step 4.3.1.10.1

Factor $x$ out of $- x$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x \cdot - 1 + 2 x i}{5} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.10.2

Factor $x$ out of $2 x i$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x \cdot - 1 + x (2 i)}{5} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.10.3

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x}{1 + 2 i}$

Step 4.3.1.11

Multiply the numerator and denominator of $\frac{2 i x}{1 + 2 i}$ by the conjugate of $1 + 2 i$ to make the denominator real.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x}{1 + 2 i} \cdot \frac{1 - 2 i}{1 - 2 i}$

Step 4.3.1.12

Multiply.

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Step 4.3.1.12.1

Combine.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x (1 - 2 i)}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2

Simplify the numerator.

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Step 4.3.1.12.2.1

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x \cdot 1 + 2 i x (- 2 i)}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2.2

Multiply $2$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 2 i x (- 2 i)}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2.3

Multiply $2 i x (- 2 i)$ .

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Step 4.3.1.12.2.3.1

Multiply $- 2$ by $2$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x - 4 i x i}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2.3.2

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x - 4 x (i^{1} i)}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2.3.3

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x - 4 x (i^{1} i^{1})}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x - 4 x i^{1 + 1}}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2.3.5

Add $1$ and $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x - 4 x i^{2}}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2.4

Simplify each term.

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Step 4.3.1.12.2.4.1

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x - 4 x \cdot - 1}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.2.4.2

Multiply $- 1$ by $- 4$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{(1 + 2 i) (1 - 2 i)}$

Step 4.3.1.12.3

Simplify the denominator.

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Step 4.3.1.12.3.1

Expand $(1 + 2 i) (1 - 2 i)$ using the FOIL Method.

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Step 4.3.1.12.3.1.1

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 (1 - 2 i) + 2 i (1 - 2 i)}$

Step 4.3.1.12.3.1.2

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 \cdot 1 + 1 (- 2 i) + 2 i (1 - 2 i)}$

Step 4.3.1.12.3.1.3

Apply the distributive property.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 \cdot 1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)}$

Step 4.3.1.12.3.2

Simplify.

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Step 4.3.1.12.3.2.1

Multiply $1$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)}$

Step 4.3.1.12.3.2.2

Multiply $- 2$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 2 i + 2 i \cdot 1 + 2 i (- 2 i)}$

Step 4.3.1.12.3.2.3

Multiply $2$ by $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 2 i + 2 i + 2 i (- 2 i)}$

Step 4.3.1.12.3.2.4

Multiply $- 2$ by $2$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 2 i + 2 i - 4 i i}$

Step 4.3.1.12.3.2.5

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 2 i + 2 i - 4 (i^{1} i)}$

Step 4.3.1.12.3.2.6

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

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 2 i + 2 i - 4 (i^{1} i^{1})}$

Step 4.3.1.12.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 2 i + 2 i - 4 i^{1 + 1}}$

Step 4.3.1.12.3.2.8

Add $1$ and $1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 2 i + 2 i - 4 i^{2}}$

Step 4.3.1.12.3.2.9

Add $- 2 i$ and $2 i$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 + 0 - 4 i^{2}}$

Step 4.3.1.12.3.2.10

Add $1$ and $0$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 4 i^{2}}$

Step 4.3.1.12.3.3

Simplify each term.

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Step 4.3.1.12.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 - 4 \cdot - 1}$

Step 4.3.1.12.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{1 + 4}$

Step 4.3.1.12.3.4

Add $1$ and $4$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 i x + 4 x}{5}$

Step 4.3.1.13

Factor $2 x$ out of $2 i x + 4 x$ .

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Step 4.3.1.13.1

Factor $2 x$ out of $2 i x$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 x (i) + 4 x}{5}$

Step 4.3.1.13.2

Factor $2 x$ out of $4 x$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 x (i) + 2 x (2)}{5}$

Step 4.3.1.13.3

Factor $2 x$ out of $2 x (i) + 2 x (2)$ .

$y = \frac{1}{5} - \frac{2 i}{5} + \frac{2}{5} + \frac{i}{5} + \frac{x (- 1 + 2 i)}{5} + \frac{2 x (i + 2)}{5}$

Step 4.3.2

Combine the numerators over the common denominator.

$y = \frac{1 - 2 i + 2 + i + x (- 1 + 2 i) + 2 x (i + 2)}{5}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$y = \frac{1 - 2 i + 2 + i + x \cdot - 1 + x (2 i) + 2 x (i + 2)}{5}$

Step 4.3.3.2

Move $- 1$ to the left of $x$ .

$y = \frac{1 - 2 i + 2 + i - 1 \cdot x + x (2 i) + 2 x (i + 2)}{5}$

Step 4.3.3.3

Simplify each term.

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Step 4.3.3.3.1

Rewrite $- 1 x$ as $- x$ .

$y = \frac{1 - 2 i + 2 + i - x + x \cdot 2 i + 2 x (i + 2)}{5}$

Step 4.3.3.3.2

Move $2$ to the left of $x$ .

$y = \frac{1 - 2 i + 2 + i - x + 2 x i + 2 x (i + 2)}{5}$

Step 4.3.3.4

Apply the distributive property.

$y = \frac{1 - 2 i + 2 + i - x + 2 x i + 2 x i + 2 x \cdot 2}{5}$

Step 4.3.3.5

Multiply $2$ by $2$ .

$y = \frac{1 - 2 i + 2 + i - x + 2 x i + 2 x i + 4 x}{5}$

Step 4.3.4

Simplify by adding terms.

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Step 4.3.4.1

Add $1$ and $2$ .

$y = \frac{3 - 2 i + i - x + 2 x i + 2 x i + 4 x}{5}$

Step 4.3.4.2

Add $- 2 i$ and $i$ .

$y = \frac{3 - i - x + 2 x i + 2 x i + 4 x}{5}$

Step 4.3.4.3

Add $- x$ and $4 x$ .

$y = \frac{3 - i + 2 x i + 2 x i + 3 x}{5}$

Step 4.3.4.4

Add $2 x i$ and $2 x i$ .

$y = \frac{3 - i + 4 x i + 3 x}{5}$

Step 4.3.4.5

Reorder terms.

$y = \frac{4 i x + 3 x + 3 - i}{5}$

Step 5

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 6