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 $x$ to the right side of the equation.

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Step 2.1

Subtract $y$ from both sides of the equation.

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

Step 2.2

Subtract $2 i y$ from both sides of the equation.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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Step 4.1

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

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

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.

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

Step 4.2.1.2

Divide $x$ by $1$ .

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

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.

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

Step 4.3.1.2

Multiply.

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Step 4.3.1.2.1

Combine.

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

Step 4.3.1.2.2

Multiply $1 + 2 i$ by $1$ .

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

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.

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

Step 4.3.1.2.3.1.2

Apply the distributive property.

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

Step 4.3.1.2.3.1.3

Apply the distributive property.

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

Step 4.3.1.2.3.2

Simplify.

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Step 4.3.1.2.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.2.3.2.2

Multiply $2$ by $1$ .

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

Step 4.3.1.2.3.2.3

Multiply $- 2$ by $1$ .

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

Step 4.3.1.2.3.2.4

Multiply $2$ by $- 2$ .

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

Step 4.3.1.2.3.2.5

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

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

Step 4.3.1.2.3.2.6

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

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

Step 4.3.1.2.3.2.7

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

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

Step 4.3.1.2.3.2.8

Add $1$ and $1$ .

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

Step 4.3.1.2.3.2.9

Subtract $2 i$ from $2 i$ .

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

Step 4.3.1.2.3.2.10

Add $1$ and $0$ .

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

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 $- 1$ .

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

Step 4.3.1.2.3.3.2

Multiply $- 4$ by $- 1$ .

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

Step 4.3.1.2.3.4

Add $1$ and $4$ .

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

Step 4.3.1.3

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

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

Step 4.3.1.4

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

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

Step 4.3.1.5

Multiply.

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Step 4.3.1.5.1

Combine.

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

Step 4.3.1.5.2

Simplify the numerator.

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Step 4.3.1.5.2.1

Apply the distributive property.

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

Step 4.3.1.5.2.2

Multiply $i$ by $1$ .

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

Step 4.3.1.5.2.3

Multiply $i (2 i)$ .

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Step 4.3.1.5.2.3.1

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

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

Step 4.3.1.5.2.3.2

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

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

Step 4.3.1.5.2.3.3

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

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

Step 4.3.1.5.2.3.4

Add $1$ and $1$ .

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

Step 4.3.1.5.2.4

Simplify each term.

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Step 4.3.1.5.2.4.1

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

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

Step 4.3.1.5.2.4.2

Multiply $2$ by $- 1$ .

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

Step 4.3.1.5.2.5

Reorder $i$ and $- 2$ .

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

Step 4.3.1.5.3

Simplify the denominator.

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Step 4.3.1.5.3.1

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

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Step 4.3.1.5.3.1.1

Apply the distributive property.

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

Step 4.3.1.5.3.1.2

Apply the distributive property.

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

Step 4.3.1.5.3.1.3

Apply the distributive property.

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

Step 4.3.1.5.3.2

Simplify.

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Step 4.3.1.5.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.5.3.2.2

Multiply $2$ by $1$ .

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

Step 4.3.1.5.3.2.3

Multiply $- 2$ by $1$ .

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

Step 4.3.1.5.3.2.4

Multiply $2$ by $- 2$ .

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

Step 4.3.1.5.3.2.5

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

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

Step 4.3.1.5.3.2.6

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

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

Step 4.3.1.5.3.2.7

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

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

Step 4.3.1.5.3.2.8

Add $1$ and $1$ .

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

Step 4.3.1.5.3.2.9

Subtract $2 i$ from $2 i$ .

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

Step 4.3.1.5.3.2.10

Add $1$ and $0$ .

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

Step 4.3.1.5.3.3

Simplify each term.

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Step 4.3.1.5.3.3.1

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

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

Step 4.3.1.5.3.3.2

Multiply $- 4$ by $- 1$ .

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

Step 4.3.1.5.3.4

Add $1$ and $4$ .

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

Step 4.3.1.6

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

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

Step 4.3.1.7

Move the negative in front of the fraction.

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

Step 4.3.1.8

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

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

Step 4.3.1.9

Multiply.

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Step 4.3.1.9.1

Combine.

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

Step 4.3.1.9.2

Simplify the numerator.

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Step 4.3.1.9.2.1

Apply the distributive property.

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

Step 4.3.1.9.2.2

Multiply $- 1$ by $1$ .

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

Step 4.3.1.9.2.3

Multiply $2$ by $- 1$ .

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

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.

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

Step 4.3.1.9.3.1.2

Apply the distributive property.

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

Step 4.3.1.9.3.1.3

Apply the distributive property.

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

Step 4.3.1.9.3.2

Simplify.

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Step 4.3.1.9.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.9.3.2.2

Multiply $2$ by $1$ .

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

Step 4.3.1.9.3.2.3

Multiply $- 2$ by $1$ .

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

Step 4.3.1.9.3.2.4

Multiply $2$ by $- 2$ .

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

Step 4.3.1.9.3.2.5

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

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

Step 4.3.1.9.3.2.6

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

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

Step 4.3.1.9.3.2.7

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

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

Step 4.3.1.9.3.2.8

Add $1$ and $1$ .

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

Step 4.3.1.9.3.2.9

Subtract $2 i$ from $2 i$ .

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

Step 4.3.1.9.3.2.10

Add $1$ and $0$ .

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

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 $- 1$ .

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

Step 4.3.1.9.3.3.2

Multiply $- 4$ by $- 1$ .

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

Step 4.3.1.9.3.4

Add $1$ and $4$ .

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

Step 4.3.1.10

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

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Step 4.3.1.10.1

Factor $y$ out of $- y$ .

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

Step 4.3.1.10.2

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

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

Step 4.3.1.10.3

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

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

Step 4.3.1.11

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

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

Step 4.3.1.12

Multiply.

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Step 4.3.1.12.1

Combine.

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

Step 4.3.1.12.2

Simplify the numerator.

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Step 4.3.1.12.2.1

Apply the distributive property.

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

Step 4.3.1.12.2.2

Multiply $- 2$ by $1$ .

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

Step 4.3.1.12.2.3

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

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Step 4.3.1.12.2.3.1

Multiply $2$ by $- 2$ .

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

Step 4.3.1.12.2.3.2

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

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

Step 4.3.1.12.2.3.3

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

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

Step 4.3.1.12.2.3.4

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

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

Step 4.3.1.12.2.3.5

Add $1$ and $1$ .

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

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 $- 1$ .

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

Step 4.3.1.12.2.4.2

Multiply $- 1$ by $- 4$ .

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

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.

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

Step 4.3.1.12.3.1.2

Apply the distributive property.

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

Step 4.3.1.12.3.1.3

Apply the distributive property.

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

Step 4.3.1.12.3.2

Simplify.

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Step 4.3.1.12.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.12.3.2.2

Multiply $2$ by $1$ .

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

Step 4.3.1.12.3.2.3

Multiply $- 2$ by $1$ .

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

Step 4.3.1.12.3.2.4

Multiply $2$ by $- 2$ .

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

Step 4.3.1.12.3.2.5

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

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

Step 4.3.1.12.3.2.6

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

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

Step 4.3.1.12.3.2.7

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

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

Step 4.3.1.12.3.2.8

Add $1$ and $1$ .

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

Step 4.3.1.12.3.2.9

Subtract $2 i$ from $2 i$ .

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

Step 4.3.1.12.3.2.10

Add $1$ and $0$ .

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

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 $- 1$ .

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

Step 4.3.1.12.3.3.2

Multiply $- 4$ by $- 1$ .

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

Step 4.3.1.12.3.4

Add $1$ and $4$ .

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

Step 4.3.1.13

Factor $2 y$ out of $- 2 i y + 4 y$ .

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Step 4.3.1.13.1

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

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

Step 4.3.1.13.2

Factor $2 y$ out of $4 y$ .

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

Step 4.3.1.13.3

Factor $2 y$ out of $2 y (- i) + 2 y (2)$ .

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

Step 4.3.2

Combine the numerators over the common denominator.

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

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

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

Step 4.3.3.3

Rewrite $- 1 y$ as $- y$ .

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

Step 4.3.3.4

Apply the distributive property.

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

Step 4.3.3.5

Multiply $- 1$ by $2$ .

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

Step 4.3.3.6

Multiply $2$ by $2$ .

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

Step 4.3.4

Simplify by adding terms.

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Step 4.3.4.1

Subtract $2$ from $1$ .

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

Step 4.3.4.2

Add $2 i$ and $i$ .

$x = \frac{- 1 + 3 i - y + y (- 2 i) - 2 y i + 4 y}{5}$

Step 4.3.4.3

Add $- y$ and $4 y$ .

$x = \frac{- 1 + 3 i + y (- 2 i) - 2 y i + 3 y}{5}$

Step 4.3.5

Subtract $2 y i$ from $y (- 2 i)$ .

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Step 4.3.5.1

Reorder $y$ and $- 2$ .

$x = \frac{- 1 + 3 i - 2 \cdot y i - 2 y i + 3 y}{5}$

Step 4.3.5.2

Subtract $2 y i$ from $- 2 \cdot y i$ .

$x = \frac{- 1 + 3 i - 4 y i + 3 y}{5}$

Step 4.3.6

Reorder terms.

$x = \frac{- 4 i y + 3 y - 1 + 3 i}{5}$

Step 4.3.7

Factor $- 1$ out of $- 4 i y$ .

$x = \frac{- (4 i y) + 3 y - 1 + 3 i}{5}$

Step 4.3.8

Factor $- 1$ out of $3 y$ .

$x = \frac{- (4 i y) - (- 3 y) - 1 + 3 i}{5}$

Step 4.3.9

Factor $- 1$ out of $- (4 i y) - (- 3 y)$ .

$x = \frac{- (4 i y - 3 y) - 1 + 3 i}{5}$

Step 4.3.10

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

$x = \frac{- (4 i y - 3 y) - 1 (1) + 3 i}{5}$

Step 4.3.11

Factor $- 1$ out of $- (4 i y - 3 y) - 1 (1)$ .

$x = \frac{- (4 i y - 3 y + 1) + 3 i}{5}$

Step 4.3.12

Factor $- 1$ out of $3 i$ .

$x = \frac{- (4 i y - 3 y + 1) - (- 3 i)}{5}$

Step 4.3.13

Factor $- 1$ out of $- (4 i y - 3 y + 1) - (- 3 i)$ .

$x = \frac{- (4 i y - 3 y + 1 - 3 i)}{5}$

Step 4.3.14

Simplify the expression.

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Step 4.3.14.1

Rewrite $- (4 i y - 3 y + 1 - 3 i)$ as $- 1 (4 i y - 3 y + 1 - 3 i)$ .

$x = \frac{- 1 (4 i y - 3 y + 1 - 3 i)}{5}$

Step 4.3.14.2

Move the negative in front of the fraction.

$x = - \frac{4 i y - 3 y + 1 - 3 i}{5}$