Trigonometry Examples

$x + 2 i - y = y i - 3 x i$

Step 1

Add $3 x i$ to both sides of the equation.

$x + 2 i - y + 3 x i = y i$

Step 2

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

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Step 2.1

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

$x - y + 3 x i = y i - 2 i$

Step 2.2

Add $y$ to both sides of the equation.

$x + 3 x i = y i - 2 i + y$

Step 3

Factor $x$ out of $x + 3 x i$ .

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Step 3.1

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

$x \cdot 1 + 3 x i = y i - 2 i + y$

Step 3.2

Factor $x$ out of $3 x i$ .

$x \cdot 1 + x (3 i) = y i - 2 i + y$

Step 3.3

Factor $x$ out of $x \cdot 1 + x (3 i)$ .

$x (1 + 3 i) = y i - 2 i + y$

Step 4

Divide each term in $x (1 + 3 i) = y i - 2 i + y$ by $1 + 3 i$ and simplify.

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Step 4.1

Divide each term in $x (1 + 3 i) = y i - 2 i + y$ by $1 + 3 i$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $1 + 3 i$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

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

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

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

Step 4.3.1.2

Multiply.

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Step 4.3.1.2.1

Combine.

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

Step 4.3.1.2.2

Simplify the numerator.

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Step 4.3.1.2.2.1

Apply the distributive property.

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

Step 4.3.1.2.2.2

Multiply $y$ by $1$ .

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

Step 4.3.1.2.2.3

Multiply $y i (- 3 i)$ .

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Step 4.3.1.2.2.3.1

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

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

Step 4.3.1.2.2.3.2

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

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

Step 4.3.1.2.2.3.3

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

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

Step 4.3.1.2.2.3.4

Add $1$ and $1$ .

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

Step 4.3.1.2.2.4

Simplify each term.

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Step 4.3.1.2.2.4.1

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

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

Step 4.3.1.2.2.4.2

Multiply $- 3$ by $- 1$ .

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

Step 4.3.1.2.2.4.3

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

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

Step 4.3.1.2.3

Simplify the denominator.

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Step 4.3.1.2.3.1

Expand $(1 + 3 i) (1 - 3 i)$ using the FOIL Method.

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Step 4.3.1.2.3.1.1

Apply the distributive property.

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

Step 4.3.1.2.3.1.2

Apply the distributive property.

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

Step 4.3.1.2.3.1.3

Apply the distributive property.

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

Step 4.3.1.2.3.2

Simplify.

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Step 4.3.1.2.3.2.1

Multiply $1$ by $1$ .

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

Step 4.3.1.2.3.2.2

Multiply $- 3$ by $1$ .

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

Step 4.3.1.2.3.2.3

Multiply $3$ by $1$ .

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

Step 4.3.1.2.3.2.4

Multiply $- 3$ by $3$ .

$x = \frac{y i + 3 y}{1 - 3 i + 3 i - 9 i i} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.2.3.2.5

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

$x = \frac{y i + 3 y}{1 - 3 i + 3 i - 9 (i^{1} i)} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.2.3.2.6

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

$x = \frac{y i + 3 y}{1 - 3 i + 3 i - 9 (i^{1} i^{1})} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.2.3.2.7

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

$x = \frac{y i + 3 y}{1 - 3 i + 3 i - 9 i^{1 + 1}} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.2.3.2.8

Add $1$ and $1$ .

$x = \frac{y i + 3 y}{1 - 3 i + 3 i - 9 i^{2}} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.2.3.2.9

Add $- 3 i$ and $3 i$ .

$x = \frac{y i + 3 y}{1 + 0 - 9 i^{2}} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.2.3.2.10

Add $1$ and $0$ .

$x = \frac{y i + 3 y}{1 - 9 i^{2}} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

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 $- 1$ .

$x = \frac{y i + 3 y}{1 - 9 \cdot - 1} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.2.3.3.2

Multiply $- 9$ by $- 1$ .

$x = \frac{y i + 3 y}{1 + 9} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.2.3.4

Add $1$ and $9$ .

$x = \frac{y i + 3 y}{10} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.3

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

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Step 4.3.1.3.1

Factor $y$ out of $y i$ .

$x = \frac{y (i) + 3 y}{10} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.3.2

Factor $y$ out of $3 y$ .

$x = \frac{y (i) + y \cdot 3}{10} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.3.3

Factor $y$ out of $y (i) + y \cdot 3$ .

$x = \frac{y (i + 3)}{10} + \frac{- 2 i}{1 + 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.4

Multiply the numerator and denominator of $\frac{- 2 i}{1 + 3 i}$ by the conjugate of $1 + 3 i$ to make the denominator real.

$x = \frac{y (i + 3)}{10} + \frac{- 2 i}{1 + 3 i} \cdot \frac{1 - 3 i}{1 - 3 i} + \frac{y}{1 + 3 i}$

Step 4.3.1.5

Multiply.

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Step 4.3.1.5.1

Combine.

$x = \frac{y (i + 3)}{10} + \frac{- 2 i (1 - 3 i)}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2

Simplify the numerator.

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Step 4.3.1.5.2.1

Apply the distributive property.

$x = \frac{y (i + 3)}{10} + \frac{- 2 i \cdot 1 - 2 i (- 3 i)}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2.2

Multiply $- 2$ by $1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 2 i - 2 i (- 3 i)}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2.3

Multiply $- 2 i (- 3 i)$ .

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Step 4.3.1.5.2.3.1

Multiply $- 3$ by $- 2$ .

$x = \frac{y (i + 3)}{10} + \frac{- 2 i + 6 i i}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2.3.2

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

$x = \frac{y (i + 3)}{10} + \frac{- 2 i + 6 (i^{1} i)}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2.3.3

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

$x = \frac{y (i + 3)}{10} + \frac{- 2 i + 6 (i^{1} i^{1})}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2.3.4

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

$x = \frac{y (i + 3)}{10} + \frac{- 2 i + 6 i^{1 + 1}}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2.3.5

Add $1$ and $1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 2 i + 6 i^{2}}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

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 $- 1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 2 i + 6 \cdot - 1}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2.4.2

Multiply $6$ by $- 1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 2 i - 6}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.2.5

Reorder $- 2 i$ and $- 6$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{(1 + 3 i) (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3

Simplify the denominator.

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Step 4.3.1.5.3.1

Expand $(1 + 3 i) (1 - 3 i)$ using the FOIL Method.

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Step 4.3.1.5.3.1.1

Apply the distributive property.

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 (1 - 3 i) + 3 i (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.1.2

Apply the distributive property.

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 \cdot 1 + 1 (- 3 i) + 3 i (1 - 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.1.3

Apply the distributive property.

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 \cdot 1 + 1 (- 3 i) + 3 i \cdot 1 + 3 i (- 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2

Simplify.

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Step 4.3.1.5.3.2.1

Multiply $1$ by $1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 + 1 (- 3 i) + 3 i \cdot 1 + 3 i (- 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.2

Multiply $- 3$ by $1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 3 i + 3 i \cdot 1 + 3 i (- 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.3

Multiply $3$ by $1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 3 i + 3 i + 3 i (- 3 i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.4

Multiply $- 3$ by $3$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 3 i + 3 i - 9 i i} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.5

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

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 3 i + 3 i - 9 (i^{1} i)} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.6

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

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 3 i + 3 i - 9 (i^{1} i^{1})} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.7

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

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 3 i + 3 i - 9 i^{1 + 1}} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.8

Add $1$ and $1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 3 i + 3 i - 9 i^{2}} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.9

Add $- 3 i$ and $3 i$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 + 0 - 9 i^{2}} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.2.10

Add $1$ and $0$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 9 i^{2}} + \frac{y}{1 + 3 i}$

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 $- 1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 - 9 \cdot - 1} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.3.2

Multiply $- 9$ by $- 1$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{1 + 9} + \frac{y}{1 + 3 i}$

Step 4.3.1.5.3.4

Add $1$ and $9$ .

$x = \frac{y (i + 3)}{10} + \frac{- 6 - 2 i}{10} + \frac{y}{1 + 3 i}$

Step 4.3.1.6

Cancel the common factor of $- 6 - 2 i$ and $10$ .

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Step 4.3.1.6.1

Factor $2$ out of $- 6$ .

$x = \frac{y (i + 3)}{10} + \frac{2 (- 3) - 2 i}{10} + \frac{y}{1 + 3 i}$

Step 4.3.1.6.2

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

$x = \frac{y (i + 3)}{10} + \frac{2 (- 3) + 2 (- i)}{10} + \frac{y}{1 + 3 i}$

Step 4.3.1.6.3

Factor $2$ out of $2 (- 3) + 2 (- i)$ .

$x = \frac{y (i + 3)}{10} + \frac{2 (- 3 - i)}{10} + \frac{y}{1 + 3 i}$

Step 4.3.1.6.4

Cancel the common factors.

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Step 4.3.1.6.4.1

Factor $2$ out of $10$ .

$x = \frac{y (i + 3)}{10} + \frac{2 (- 3 - i)}{2 \cdot 5} + \frac{y}{1 + 3 i}$

Step 4.3.1.6.4.2

Cancel the common factor.

$x = \frac{y (i + 3)}{10} + \frac{2 (- 3 - i)}{2 \cdot 5} + \frac{y}{1 + 3 i}$

Step 4.3.1.6.4.3

Rewrite the expression.

$x = \frac{y (i + 3)}{10} + \frac{- 3 - i}{5} + \frac{y}{1 + 3 i}$

Step 4.3.1.7

Split the fraction $\frac{- 3 - i}{5}$ into two fractions.

$x = \frac{y (i + 3)}{10} + \frac{- 3}{5} + \frac{- i}{5} + \frac{y}{1 + 3 i}$

Step 4.3.1.8

Simplify each term.

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Step 4.3.1.8.1

Move the negative in front of the fraction.

$x = \frac{y (i + 3)}{10} - \frac{3}{5} + \frac{- i}{5} + \frac{y}{1 + 3 i}$

Step 4.3.1.8.2

Move the negative in front of the fraction.

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y}{1 + 3 i}$

Step 4.3.1.9

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y}{1 + 3 i} \cdot \frac{1 - 3 i}{1 - 3 i}$

Step 4.3.1.10

Multiply.

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Step 4.3.1.10.1

Combine.

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y (1 - 3 i)}{(1 + 3 i) (1 - 3 i)}$

Step 4.3.1.10.2

Simplify the numerator.

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Step 4.3.1.10.2.1

Apply the distributive property.

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y \cdot 1 + y (- 3 i)}{(1 + 3 i) (1 - 3 i)}$

Step 4.3.1.10.2.2

Multiply $y$ by $1$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y + y (- 3 i)}{(1 + 3 i) (1 - 3 i)}$

Step 4.3.1.10.2.3

Reorder factors in $y + y (- 3 i)$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{(1 + 3 i) (1 - 3 i)}$

Step 4.3.1.10.3

Simplify the denominator.

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Step 4.3.1.10.3.1

Expand $(1 + 3 i) (1 - 3 i)$ using the FOIL Method.

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Step 4.3.1.10.3.1.1

Apply the distributive property.

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 (1 - 3 i) + 3 i (1 - 3 i)}$

Step 4.3.1.10.3.1.2

Apply the distributive property.

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 \cdot 1 + 1 (- 3 i) + 3 i (1 - 3 i)}$

Step 4.3.1.10.3.1.3

Apply the distributive property.

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 \cdot 1 + 1 (- 3 i) + 3 i \cdot 1 + 3 i (- 3 i)}$

Step 4.3.1.10.3.2

Simplify.

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Step 4.3.1.10.3.2.1

Multiply $1$ by $1$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 + 1 (- 3 i) + 3 i \cdot 1 + 3 i (- 3 i)}$

Step 4.3.1.10.3.2.2

Multiply $- 3$ by $1$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 3 i + 3 i \cdot 1 + 3 i (- 3 i)}$

Step 4.3.1.10.3.2.3

Multiply $3$ by $1$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 3 i + 3 i + 3 i (- 3 i)}$

Step 4.3.1.10.3.2.4

Multiply $- 3$ by $3$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 3 i + 3 i - 9 i i}$

Step 4.3.1.10.3.2.5

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 3 i + 3 i - 9 (i^{1} i)}$

Step 4.3.1.10.3.2.6

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 3 i + 3 i - 9 (i^{1} i^{1})}$

Step 4.3.1.10.3.2.7

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 3 i + 3 i - 9 i^{1 + 1}}$

Step 4.3.1.10.3.2.8

Add $1$ and $1$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 3 i + 3 i - 9 i^{2}}$

Step 4.3.1.10.3.2.9

Add $- 3 i$ and $3 i$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 + 0 - 9 i^{2}}$

Step 4.3.1.10.3.2.10

Add $1$ and $0$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 9 i^{2}}$

Step 4.3.1.10.3.3

Simplify each term.

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Step 4.3.1.10.3.3.1

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 - 9 \cdot - 1}$

Step 4.3.1.10.3.3.2

Multiply $- 9$ by $- 1$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{1 + 9}$

Step 4.3.1.10.3.4

Add $1$ and $9$ .

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y - 3 y i}{10}$

Step 4.3.1.11

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

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Step 4.3.1.11.1

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y^{1} - 3 y i}{10}$

Step 4.3.1.11.2

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y \cdot 1 - 3 y i}{10}$

Step 4.3.1.11.3

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y \cdot 1 + y (- 3 i)}{10}$

Step 4.3.1.11.4

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

$x = \frac{y (i + 3)}{10} - \frac{3}{5} - \frac{i}{5} + \frac{y (1 - 3 i)}{10}$

Step 4.3.2

Combine the numerators over the common denominator.

$x = \frac{y (i + 3) + y (1 - 3 i)}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$x = \frac{y i + y \cdot 3 + y (1 - 3 i)}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.3.2

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

$x = \frac{y i + 3 \cdot y + y (1 - 3 i)}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.3.3

Apply the distributive property.

$x = \frac{y i + 3 y + y \cdot 1 + y (- 3 i)}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.3.4

Multiply $y$ by $1$ .

$x = \frac{y i + 3 y + y + y (- 3 i)}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.4

Add $y i$ and $y (- 3 i)$ .

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Step 4.3.4.1

Reorder $y$ and $- 3$ .

$x = \frac{y i - 3 \cdot y i + 3 y + y}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.4.2

Subtract $3 \cdot y i$ from $y i$ .

$x = \frac{- 2 y i + 3 y + y}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.5

Simplify terms.

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Step 4.3.5.1

Add $3 y$ and $y$ .

$x = \frac{- 2 y i + 4 y}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.5.2

Simplify each term.

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Step 4.3.5.2.1

Simplify the numerator.

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Step 4.3.5.2.1.1

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

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Step 4.3.5.2.1.1.1

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

$x = \frac{2 y (- 1 i) + 4 y}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.5.2.1.1.2

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

$x = \frac{2 y (- 1 i) + 2 y (2)}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.5.2.1.1.3

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

$x = \frac{2 y (- 1 i + 2)}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.5.2.1.2

Rewrite $- 1 i$ as $- i$ .

$x = \frac{2 y (- i + 2)}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.5.2.2

Cancel the common factor of $2$ and $10$ .

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Step 4.3.5.2.2.1

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

$x = \frac{2 (y (- i + 2))}{10} + \frac{- 3}{5} + \frac{- i}{5}$

Step 4.3.5.2.2.2

Cancel the common factors.

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Step 4.3.5.2.2.2.1

Factor $2$ out of $10$ .

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

Step 4.3.5.2.2.2.2

Cancel the common factor.

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

Step 4.3.5.2.2.2.3

Rewrite the expression.

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

Step 4.3.5.2.3

Move the negative in front of the fraction.

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

Step 4.3.5.2.4

Move the negative in front of the fraction.

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

Step 4.3.5.3

Combine the numerators over the common denominator.

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

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Apply the distributive property.

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

Step 4.3.6.2

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

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

Step 4.3.7

Simplify terms.

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Step 4.3.7.1

Combine the numerators over the common denominator.

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

Step 4.3.7.2

Reorder factors in $y (- i) + 2 y - 3 - i$ .

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

Step 4.3.7.3

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

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

Step 4.3.7.4

Factor $- 1$ out of $2 y$ .

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

Step 4.3.7.5

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

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

Step 4.3.7.6

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 4.3.7.7

Factor $- 1$ out of $- (y i - 2 y) - 1 (3)$ .

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

Step 4.3.7.8

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

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

Step 4.3.7.9

Factor $- 1$ out of $- (y i - 2 y + 3) - (i)$ .

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

Step 4.3.7.10

Simplify the expression.

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Step 4.3.7.10.1

Rewrite $- (y i - 2 y + 3 + i)$ as $- 1 (y i - 2 y + 3 + i)$ .

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

Step 4.3.7.10.2

Move the negative in front of the fraction.

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