Finite Math Examples

$(1 + i) x + (3 + 5 i) y = 1 + 3 i$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$1 x + i x + (3 + 5 i) y = 1 + 3 i$

Step 1.2

Multiply $x$ by $1$ .

$x + i x + (3 + 5 i) y = 1 + 3 i$

Step 1.3

Apply the distributive property.

$x + i x + 3 y + 5 i y = 1 + 3 i$

Step 2

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

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Step 2.1

Subtract $3 y$ from both sides of the equation.

$x + i x + 5 i y = 1 + 3 i - 3 y$

Step 2.2

Subtract $5 i y$ from both sides of the equation.

$x + i x = 1 + 3 i - 3 y - 5 i y$

Step 3

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

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Step 3.1

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

$x \cdot 1 + i x = 1 + 3 i - 3 y - 5 i y$

Step 3.2

Factor $x$ out of $i x$ .

$x \cdot 1 + x i = 1 + 3 i - 3 y - 5 i y$

Step 3.3

Factor $x$ out of $x \cdot 1 + x i$ .

$x (1 + i) = 1 + 3 i - 3 y - 5 i y$

Step 4

Divide each term in $x (1 + i) = 1 + 3 i - 3 y - 5 i y$ by $1 + i$ and simplify.

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Step 4.1

Divide each term in $x (1 + i) = 1 + 3 i - 3 y - 5 i y$ by $1 + i$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $1 + i$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

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

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

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

Step 4.3.1.2

Multiply.

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Step 4.3.1.2.1

Combine.

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

Step 4.3.1.2.2

Multiply $1 - i$ by $1$ .

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

Step 4.3.1.2.3

Simplify the denominator.

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Step 4.3.1.2.3.1

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

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Step 4.3.1.2.3.1.1

Apply the distributive property.

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

Step 4.3.1.2.3.1.2

Apply the distributive property.

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

Step 4.3.1.2.3.1.3

Apply the distributive property.

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

Step 4.3.1.2.3.2

Simplify.

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Step 4.3.1.2.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.2.3.2.2

Multiply $- 1$ by $1$ .

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

Step 4.3.1.2.3.2.3

Multiply $1$ by $1$ .

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

Step 4.3.1.2.3.2.4

Multiply $- 1$ by $1$ .

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

Step 4.3.1.2.3.2.5

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

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

Step 4.3.1.2.3.2.6

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

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

Step 4.3.1.2.3.2.7

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

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

Step 4.3.1.2.3.2.8

Add $1$ and $1$ .

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

Step 4.3.1.2.3.2.9

Add $- 1 i$ and $1 i$ .

$x = \frac{1 - i}{1 + 0 - i^{2}} + \frac{3 i}{1 + i} + \frac{- 3 y}{1 + i} + \frac{- 5 i y}{1 + i}$

Step 4.3.1.2.3.2.10

Add $1$ and $0$ .

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

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 $- 1$ .

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

Step 4.3.1.2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.2.3.4

Add $1$ and $1$ .

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

Step 4.3.1.3

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

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

Step 4.3.1.4

Move the negative in front of the fraction.

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

Step 4.3.1.5

Multiply the numerator and denominator of $\frac{3 i}{1 + 1 i}$ by the conjugate of $1 + 1 i$ to make the denominator real.

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

Step 4.3.1.6

Multiply.

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Step 4.3.1.6.1

Combine.

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

Step 4.3.1.6.2

Simplify the numerator.

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Step 4.3.1.6.2.1

Apply the distributive property.

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

Step 4.3.1.6.2.2

Multiply $3$ by $1$ .

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

Step 4.3.1.6.2.3

Multiply $3 i (- i)$ .

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Step 4.3.1.6.2.3.1

Multiply $- 1$ by $3$ .

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

Step 4.3.1.6.2.3.2

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

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

Step 4.3.1.6.2.3.3

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

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

Step 4.3.1.6.2.3.4

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

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

Step 4.3.1.6.2.3.5

Add $1$ and $1$ .

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

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 $- 1$ .

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

Step 4.3.1.6.2.4.2

Multiply $- 3$ by $- 1$ .

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

Step 4.3.1.6.2.5

Reorder $3 i$ and $3$ .

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

Step 4.3.1.6.3

Simplify the denominator.

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Step 4.3.1.6.3.1

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

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Step 4.3.1.6.3.1.1

Apply the distributive property.

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

Step 4.3.1.6.3.1.2

Apply the distributive property.

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

Step 4.3.1.6.3.1.3

Apply the distributive property.

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

Step 4.3.1.6.3.2

Simplify.

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Step 4.3.1.6.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.6.3.2.2

Multiply $- 1$ by $1$ .

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

Step 4.3.1.6.3.2.3

Multiply $1$ by $1$ .

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

Step 4.3.1.6.3.2.4

Multiply $- 1$ by $1$ .

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

Step 4.3.1.6.3.2.5

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

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

Step 4.3.1.6.3.2.6

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

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

Step 4.3.1.6.3.2.7

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

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

Step 4.3.1.6.3.2.8

Add $1$ and $1$ .

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

Step 4.3.1.6.3.2.9

Add $- 1 i$ and $1 i$ .

$x = \frac{1}{2} - \frac{i}{2} + \frac{3 + 3 i}{1 + 0 - i^{2}} + \frac{- 3 y}{1 + i} + \frac{- 5 i y}{1 + i}$

Step 4.3.1.6.3.2.10

Add $1$ and $0$ .

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

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 $- 1$ .

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

Step 4.3.1.6.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.6.3.4

Add $1$ and $1$ .

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

Step 4.3.1.7

Split the fraction $\frac{3 + 3 i}{2}$ into two fractions.

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

Step 4.3.1.8

Multiply the numerator and denominator of $\frac{- 3 y}{1 + 1 i}$ by the conjugate of $1 + 1 i$ to make the denominator real.

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

Step 4.3.1.9

Multiply.

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Step 4.3.1.9.1

Combine.

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

Step 4.3.1.9.2

Simplify the numerator.

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Step 4.3.1.9.2.1

Apply the distributive property.

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

Step 4.3.1.9.2.2

Multiply $- 3$ by $1$ .

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

Step 4.3.1.9.2.3

Multiply $- 1$ by $- 3$ .

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

Step 4.3.1.9.3

Simplify the denominator.

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Step 4.3.1.9.3.1

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

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Step 4.3.1.9.3.1.1

Apply the distributive property.

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

Step 4.3.1.9.3.1.2

Apply the distributive property.

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

Step 4.3.1.9.3.1.3

Apply the distributive property.

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

Step 4.3.1.9.3.2

Simplify.

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Step 4.3.1.9.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.9.3.2.2

Multiply $- 1$ by $1$ .

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

Step 4.3.1.9.3.2.3

Multiply $1$ by $1$ .

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

Step 4.3.1.9.3.2.4

Multiply $- 1$ by $1$ .

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

Step 4.3.1.9.3.2.5

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

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

Step 4.3.1.9.3.2.6

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

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

Step 4.3.1.9.3.2.7

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

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

Step 4.3.1.9.3.2.8

Add $1$ and $1$ .

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

Step 4.3.1.9.3.2.9

Add $- 1 i$ and $1 i$ .

$x = \frac{1}{2} - \frac{i}{2} + \frac{3}{2} + \frac{3 i}{2} + \frac{- 3 y + 3 y i}{1 + 0 - i^{2}} + \frac{- 5 i y}{1 + i}$

Step 4.3.1.9.3.2.10

Add $1$ and $0$ .

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

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 $- 1$ .

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

Step 4.3.1.9.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.9.3.4

Add $1$ and $1$ .

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

Step 4.3.1.10

Factor $3 y$ out of $- 3 y + 3 y i$ .

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Step 4.3.1.10.1

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

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

Step 4.3.1.10.2

Factor $3 y$ out of $3 y i$ .

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

Step 4.3.1.10.3

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

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

Step 4.3.1.11

Multiply the numerator and denominator of $\frac{- 5 i y}{1 + 1 i}$ by the conjugate of $1 + 1 i$ to make the denominator real.

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

Step 4.3.1.12

Multiply.

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Step 4.3.1.12.1

Combine.

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

Step 4.3.1.12.2

Simplify the numerator.

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Step 4.3.1.12.2.1

Apply the distributive property.

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

Step 4.3.1.12.2.2

Multiply $- 5$ by $1$ .

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

Step 4.3.1.12.2.3

Multiply $- 5 i y (- i)$ .

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Step 4.3.1.12.2.3.1

Multiply $- 1$ by $- 5$ .

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

Step 4.3.1.12.2.3.2

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

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

Step 4.3.1.12.2.3.3

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

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

Step 4.3.1.12.2.3.4

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

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

Step 4.3.1.12.2.3.5

Add $1$ and $1$ .

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

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 $- 1$ .

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

Step 4.3.1.12.2.4.2

Multiply $- 1$ by $5$ .

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

Step 4.3.1.12.3

Simplify the denominator.

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Step 4.3.1.12.3.1

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

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Step 4.3.1.12.3.1.1

Apply the distributive property.

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

Step 4.3.1.12.3.1.2

Apply the distributive property.

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

Step 4.3.1.12.3.1.3

Apply the distributive property.

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

Step 4.3.1.12.3.2

Simplify.

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Step 4.3.1.12.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.12.3.2.2

Multiply $- 1$ by $1$ .

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

Step 4.3.1.12.3.2.3

Multiply $1$ by $1$ .

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

Step 4.3.1.12.3.2.4

Multiply $- 1$ by $1$ .

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

Step 4.3.1.12.3.2.5

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

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

Step 4.3.1.12.3.2.6

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

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

Step 4.3.1.12.3.2.7

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

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

Step 4.3.1.12.3.2.8

Add $1$ and $1$ .

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

Step 4.3.1.12.3.2.9

Add $- 1 i$ and $1 i$ .

$x = \frac{1}{2} - \frac{i}{2} + \frac{3}{2} + \frac{3 i}{2} + \frac{3 y (- 1 + i)}{2} + \frac{- 5 i y - 5 y}{1 + 0 - i^{2}}$

Step 4.3.1.12.3.2.10

Add $1$ and $0$ .

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

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 $- 1$ .

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

Step 4.3.1.12.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.12.3.4

Add $1$ and $1$ .

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

Step 4.3.1.13

Factor $5 y$ out of $- 5 i y - 5 y$ .

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Step 4.3.1.13.1

Factor $5 y$ out of $- 5 i y$ .

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

Step 4.3.1.13.2

Factor $5 y$ out of $- 5 y$ .

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

Step 4.3.1.13.3

Factor $5 y$ out of $5 y (- i) + 5 y (- 1)$ .

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

Step 4.3.2

Combine the numerators over the common denominator.

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Multiply $- 1$ by $3$ .

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

Step 4.3.3.3

Apply the distributive property.

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

Step 4.3.3.4

Multiply $- 1$ by $5$ .

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

Step 4.3.3.5

Multiply $- 1$ by $5$ .

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

Step 4.3.4

Simplify terms.

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Step 4.3.4.1

Add $1$ and $3$ .

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

Step 4.3.4.2

Add $- i$ and $3 i$ .

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

Step 4.3.4.3

Subtract $5 y$ from $- 3 y$ .

$x = \frac{4 + 2 i + 3 y i - 5 y i - 8 y}{2}$

Step 4.3.4.4

Subtract $5 y i$ from $3 y i$ .

$x = \frac{4 + 2 i - 2 y i - 8 y}{2}$

Step 4.3.4.5

Cancel the common factor of $4 + 2 i - 2 y i - 8 y$ and $2$ .

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Step 4.3.4.5.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 2 + 2 i - 2 y i - 8 y}{2}$

Step 4.3.4.5.2

Factor $2$ out of $2 i$ .

$x = \frac{2 \cdot 2 + 2 (i) - 2 y i - 8 y}{2}$

Step 4.3.4.5.3

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

$x = \frac{2 (2 + i) - 2 y i - 8 y}{2}$

Step 4.3.4.5.4

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

$x = \frac{2 (2 + i) + 2 (- y i) - 8 y}{2}$

Step 4.3.4.5.5

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

$x = \frac{2 (2 + i - y i) - 8 y}{2}$

Step 4.3.4.5.6

Factor $2$ out of $- 8 y$ .

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

Step 4.3.4.5.7

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

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

Step 4.3.4.5.8

Cancel the common factors.

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Step 4.3.4.5.8.1

Factor $2$ out of $2$ .

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

Step 4.3.4.5.8.2

Cancel the common factor.

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

Step 4.3.4.5.8.3

Rewrite the expression.

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

Step 4.3.4.5.8.4

Divide $2 + i - y i - 4 y$ by $1$ .

$x = 2 + i - y i - 4 y$