Basic Math Examples

$3 x + i x - 2 y = 12 - i y - i$

Step 1

Add $i y$ to both sides of the equation.

$3 x + i x - 2 y + i y = 12 - i$

Step 2

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

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Step 2.1

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

$i x - 2 y + i y = 12 - i - 3 x$

Step 2.2

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

$- 2 y + i y = 12 - i - 3 x - i x$

Step 3

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

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Step 3.1

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

$y \cdot - 2 + i y = 12 - i - 3 x - i x$

Step 3.2

Factor $y$ out of $i y$ .

$y \cdot - 2 + y i = 12 - i - 3 x - i x$

Step 3.3

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

$y (- 2 + i) = 12 - i - 3 x - i x$

Step 4

Divide each term in $y (- 2 + i) = 12 - i - 3 x - i x$ by $- 2 + i$ and simplify.

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Step 4.1

Divide each term in $y (- 2 + i) = 12 - i - 3 x - i x$ by $- 2 + i$ .

$\frac{y (- 2 + i)}{- 2 + i} = \frac{12}{- 2 + i} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Cancel the common factor.

$\frac{y (- 2 + i)}{- 2 + i} = \frac{12}{- 2 + i} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{- 2 + i} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

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 $\frac{12}{- 2 + 1 i}$ by the conjugate of $- 2 + 1 i$ to make the denominator real.

$y = \frac{12}{- 2 + 1 i} \cdot \frac{- 2 - i}{- 2 - i} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2

Multiply.

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Step 4.3.1.2.1

Combine.

$y = \frac{12 (- 2 - i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.2

Simplify the numerator.

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Step 4.3.1.2.2.1

Apply the distributive property.

$y = \frac{12 \cdot - 2 + 12 (- i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.2.2

Multiply $12$ by $- 2$ .

$y = \frac{- 24 + 12 (- i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.2.3

Multiply $- 1$ by $12$ .

$y = \frac{- 24 - 12 i}{(- 2 + 1 i) (- 2 - i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3

Simplify the denominator.

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Step 4.3.1.2.3.1

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

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Step 4.3.1.2.3.1.1

Apply the distributive property.

$y = \frac{- 24 - 12 i}{- 2 (- 2 - i) + 1 i (- 2 - i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.1.2

Apply the distributive property.

$y = \frac{- 24 - 12 i}{- 2 \cdot - 2 - 2 (- i) + 1 i (- 2 - i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.1.3

Apply the distributive property.

$y = \frac{- 24 - 12 i}{- 2 \cdot - 2 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2

Simplify.

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Step 4.3.1.2.3.2.1

Multiply $- 2$ by $- 2$ .

$y = \frac{- 24 - 12 i}{4 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.2

Multiply $- 1$ by $- 2$ .

$y = \frac{- 24 - 12 i}{4 + 2 i + 1 i \cdot - 2 + 1 i (- i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.3

Multiply $- 2$ by $1$ .

$y = \frac{- 24 - 12 i}{4 + 2 i - 2 i + 1 i (- i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.4

Multiply $- 1$ by $1$ .

$y = \frac{- 24 - 12 i}{4 + 2 i - 2 i - i i} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.5

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

$y = \frac{- 24 - 12 i}{4 + 2 i - 2 i - (i^{1} i)} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.6

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

$y = \frac{- 24 - 12 i}{4 + 2 i - 2 i - (i^{1} i^{1})} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.7

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

$y = \frac{- 24 - 12 i}{4 + 2 i - 2 i - i^{1 + 1}} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.8

Add $1$ and $1$ .

$y = \frac{- 24 - 12 i}{4 + 2 i - 2 i - i^{2}} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.9

Subtract $2 i$ from $2 i$ .

$y = \frac{- 24 - 12 i}{4 + 0 - i^{2}} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.2.10

Add $4$ and $0$ .

$y = \frac{- 24 - 12 i}{4 - i^{2}} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

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 $- 1$ .

$y = \frac{- 24 - 12 i}{4 - - 1} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.3.2

Multiply $- 1$ by $- 1$ .

$y = \frac{- 24 - 12 i}{4 + 1} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.2.3.4

Add $4$ and $1$ .

$y = \frac{- 24 - 12 i}{5} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.3

Split the fraction $\frac{- 24 - 12 i}{5}$ into two fractions.

$y = \frac{- 24}{5} + \frac{- 12 i}{5} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.4

Simplify each term.

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Step 4.3.1.4.1

Move the negative in front of the fraction.

$y = - \frac{24}{5} + \frac{- 12 i}{5} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.4.2

Move the negative in front of the fraction.

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- i}{- 2 + i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.5

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- i}{- 2 + 1 i} \cdot \frac{- 2 - i}{- 2 - i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6

Multiply.

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Step 4.3.1.6.1

Combine.

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- i (- 2 - i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2

Simplify the numerator.

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Step 4.3.1.6.2.1

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- i \cdot - 2 - i (- i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.2

Multiply $- 2$ by $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{2 i - i (- i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.3

Multiply $- i (- i)$ .

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Step 4.3.1.6.2.3.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{2 i + 1 i i}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.3.2

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{2 i + 1 (i^{1} i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.3.3

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{2 i + 1 (i^{1} i^{1})}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.3.4

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{2 i + 1 i^{1 + 1}}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.3.5

Add $1$ and $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{2 i + 1 i^{2}}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.3.6

Multiply $i^{2}$ by $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{2 i + i^{2}}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.4

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{2 i - 1}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.2.5

Reorder $2 i$ and $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{(- 2 + 1 i) (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3

Simplify the denominator.

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Step 4.3.1.6.3.1

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

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Step 4.3.1.6.3.1.1

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{- 2 (- 2 - i) + 1 i (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.1.2

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{- 2 \cdot - 2 - 2 (- i) + 1 i (- 2 - i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.1.3

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{- 2 \cdot - 2 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2

Simplify.

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Step 4.3.1.6.3.2.1

Multiply $- 2$ by $- 2$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.2

Multiply $- 1$ by $- 2$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 2 i + 1 i \cdot - 2 + 1 i (- i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.3

Multiply $- 2$ by $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 2 i - 2 i + 1 i (- i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.4

Multiply $- 1$ by $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 2 i - 2 i - i i} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.5

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 2 i - 2 i - (i^{1} i)} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.6

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 2 i - 2 i - (i^{1} i^{1})} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.7

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 2 i - 2 i - i^{1 + 1}} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.8

Add $1$ and $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 2 i - 2 i - i^{2}} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.9

Subtract $2 i$ from $2 i$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 0 - i^{2}} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.2.10

Add $4$ and $0$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 - i^{2}} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

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 $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 - - 1} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.3.2

Multiply $- 1$ by $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{4 + 1} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.6.3.4

Add $4$ and $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1 + 2 i}{5} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.7

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

$y = - \frac{24}{5} - \frac{12 i}{5} + \frac{- 1}{5} + \frac{2 i}{5} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.8

Move the negative in front of the fraction.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{- 3 x}{- 2 + i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.9

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{- 3 x}{- 2 + 1 i} \cdot \frac{- 2 - i}{- 2 - i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10

Multiply.

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Step 4.3.1.10.1

Combine.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{- 3 x (- 2 - i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.2

Simplify the numerator.

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Step 4.3.1.10.2.1

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{- 3 x \cdot - 2 - 3 x (- i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.2.2

Multiply $- 2$ by $- 3$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x - 3 x (- i)}{(- 2 + 1 i) (- 2 - i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.2.3

Multiply $- 1$ by $- 3$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{(- 2 + 1 i) (- 2 - i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3

Simplify the denominator.

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Step 4.3.1.10.3.1

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

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Step 4.3.1.10.3.1.1

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{- 2 (- 2 - i) + 1 i (- 2 - i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.1.2

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{- 2 \cdot - 2 - 2 (- i) + 1 i (- 2 - i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.1.3

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{- 2 \cdot - 2 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2

Simplify.

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Step 4.3.1.10.3.2.1

Multiply $- 2$ by $- 2$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.2

Multiply $- 1$ by $- 2$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 2 i + 1 i \cdot - 2 + 1 i (- i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.3

Multiply $- 2$ by $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 2 i - 2 i + 1 i (- i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.4

Multiply $- 1$ by $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 2 i - 2 i - i i} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.5

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 2 i - 2 i - (i^{1} i)} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.6

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 2 i - 2 i - (i^{1} i^{1})} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.7

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 2 i - 2 i - i^{1 + 1}} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.8

Add $1$ and $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 2 i - 2 i - i^{2}} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.9

Subtract $2 i$ from $2 i$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 0 - i^{2}} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.2.10

Add $4$ and $0$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 - i^{2}} + \frac{- i x}{- 2 + i}$

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 $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 - - 1} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.3.2

Multiply $- 1$ by $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{4 + 1} + \frac{- i x}{- 2 + i}$

Step 4.3.1.10.3.4

Add $4$ and $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{6 x + 3 x i}{5} + \frac{- i x}{- 2 + i}$

Step 4.3.1.11

Factor $3 x$ out of $6 x + 3 x i$ .

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Step 4.3.1.11.1

Factor $3 x$ out of $6 x$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2) + 3 x i}{5} + \frac{- i x}{- 2 + i}$

Step 4.3.1.11.2

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2) + 3 x (i)}{5} + \frac{- i x}{- 2 + i}$

Step 4.3.1.11.3

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{- i x}{- 2 + i}$

Step 4.3.1.12

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{- i x}{- 2 + 1 i} \cdot \frac{- 2 - i}{- 2 - i}$

Step 4.3.1.13

Multiply.

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Step 4.3.1.13.1

Combine.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{- i x (- 2 - i)}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2

Simplify the numerator.

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Step 4.3.1.13.2.1

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{- i x \cdot - 2 - i x (- i)}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.2

Multiply $- 2$ by $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - i x (- i)}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.3

Multiply $- i x (- i)$ .

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Step 4.3.1.13.2.3.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x + 1 i x i}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.3.2

Multiply $x$ by $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x + x i i}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.3.3

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x + x (i^{1} i)}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.3.4

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x + x (i^{1} i^{1})}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.3.5

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x + x i^{1 + 1}}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.3.6

Add $1$ and $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x + x i^{2}}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.4

Simplify each term.

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Step 4.3.1.13.2.4.1

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x + x \cdot - 1}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.4.2

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - 1 \cdot x}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.2.4.3

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{(- 2 + 1 i) (- 2 - i)}$

Step 4.3.1.13.3

Simplify the denominator.

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Step 4.3.1.13.3.1

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

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Step 4.3.1.13.3.1.1

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{- 2 (- 2 - i) + 1 i (- 2 - i)}$

Step 4.3.1.13.3.1.2

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{- 2 \cdot - 2 - 2 (- i) + 1 i (- 2 - i)}$

Step 4.3.1.13.3.1.3

Apply the distributive property.

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{- 2 \cdot - 2 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)}$

Step 4.3.1.13.3.2

Simplify.

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Step 4.3.1.13.3.2.1

Multiply $- 2$ by $- 2$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 - 2 (- i) + 1 i \cdot - 2 + 1 i (- i)}$

Step 4.3.1.13.3.2.2

Multiply $- 1$ by $- 2$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 2 i + 1 i \cdot - 2 + 1 i (- i)}$

Step 4.3.1.13.3.2.3

Multiply $- 2$ by $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 2 i - 2 i + 1 i (- i)}$

Step 4.3.1.13.3.2.4

Multiply $- 1$ by $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 2 i - 2 i - i i}$

Step 4.3.1.13.3.2.5

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 2 i - 2 i - (i^{1} i)}$

Step 4.3.1.13.3.2.6

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 2 i - 2 i - (i^{1} i^{1})}$

Step 4.3.1.13.3.2.7

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 2 i - 2 i - i^{1 + 1}}$

Step 4.3.1.13.3.2.8

Add $1$ and $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 2 i - 2 i - i^{2}}$

Step 4.3.1.13.3.2.9

Subtract $2 i$ from $2 i$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 0 - i^{2}}$

Step 4.3.1.13.3.2.10

Add $4$ and $0$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 - i^{2}}$

Step 4.3.1.13.3.3

Simplify each term.

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Step 4.3.1.13.3.3.1

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 - - 1}$

Step 4.3.1.13.3.3.2

Multiply $- 1$ by $- 1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{4 + 1}$

Step 4.3.1.13.3.4

Add $4$ and $1$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{2 i x - x}{5}$

Step 4.3.1.14

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

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Step 4.3.1.14.1

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{x (2 i) - x}{5}$

Step 4.3.1.14.2

Factor $x$ out of $- x$ .

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{x (2 i) + x \cdot - 1}{5}$

Step 4.3.1.14.3

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

$y = - \frac{24}{5} - \frac{12 i}{5} - \frac{1}{5} + \frac{2 i}{5} + \frac{3 x (2 + i)}{5} + \frac{x (2 i - 1)}{5}$

Step 4.3.2

Combine the numerators over the common denominator.

$y = \frac{- 24 - 12 i - 1 + 2 i + 3 x (2 + i) + x (2 i - 1)}{5}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$y = \frac{- 24 - 12 i - 1 + 2 i + 3 x \cdot 2 + 3 x i + x (2 i - 1)}{5}$

Step 4.3.3.2

Multiply $2$ by $3$ .

$y = \frac{- 24 - 12 i - 1 + 2 i + 6 x + 3 x i + x (2 i - 1)}{5}$

Step 4.3.3.3

Apply the distributive property.

$y = \frac{- 24 - 12 i - 1 + 2 i + 6 x + 3 x i + x (2 i) + x \cdot - 1}{5}$

Step 4.3.3.4

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

$y = \frac{- 24 - 12 i - 1 + 2 i + 6 x + 3 x i + x \cdot 2 i - 1 \cdot x}{5}$

Step 4.3.3.5

Simplify each term.

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Step 4.3.3.5.1

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

$y = \frac{- 24 - 12 i - 1 + 2 i + 6 x + 3 x i + 2 \cdot x i - 1 \cdot x}{5}$

Step 4.3.3.5.2

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

$y = \frac{- 24 - 12 i - 1 + 2 i + 6 x + 3 x i + 2 x i - x}{5}$

Step 4.3.4

Simplify terms.

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Step 4.3.4.1

Subtract $1$ from $- 24$ .

$y = \frac{- 25 - 12 i + 2 i + 6 x + 3 x i + 2 x i - x}{5}$

Step 4.3.4.2

Add $- 12 i$ and $2 i$ .

$y = \frac{- 25 - 10 i + 6 x + 3 x i + 2 x i - x}{5}$

Step 4.3.4.3

Subtract $x$ from $6 x$ .

$y = \frac{- 25 - 10 i + 3 x i + 2 x i + 5 x}{5}$

Step 4.3.4.4

Add $3 x i$ and $2 x i$ .

$y = \frac{- 25 - 10 i + 5 x i + 5 x}{5}$

Step 4.3.4.5

Cancel the common factor of $- 25 - 10 i + 5 x i + 5 x$ and $5$ .

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Step 4.3.4.5.1

Factor $5$ out of $- 25$ .

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

Step 4.3.4.5.2

Factor $5$ out of $- 10 i$ .

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

Step 4.3.4.5.3

Factor $5$ out of $5 (- 5) + 5 (- 2 i)$ .

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

Step 4.3.4.5.4

Factor $5$ out of $5 x i$ .

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

Step 4.3.4.5.5

Factor $5$ out of $5 (- 5 - 2 i) + 5 (x i)$ .

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

Step 4.3.4.5.6

Factor $5$ out of $5 x$ .

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

Step 4.3.4.5.7

Factor $5$ out of $5 (- 5 - 2 i + x i) + 5 (x)$ .

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

Step 4.3.4.5.8

Cancel the common factors.

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Step 4.3.4.5.8.1

Factor $5$ out of $5$ .

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

Step 4.3.4.5.8.2

Cancel the common factor.

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

Step 4.3.4.5.8.3

Rewrite the expression.

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

Step 4.3.4.5.8.4

Divide $- 5 - 2 i + x i + x$ by $1$ .

$y = - 5 - 2 i + x i + x$