Basic Math Examples

$3 x + 2 i y + i x + 5 y = 7 + 5 i$

Step 1

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

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Step 1.1

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

$2 i y + i x + 5 y = 7 + 5 i - 3 x$

Step 1.2

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

$2 i y + 5 y = 7 + 5 i - 3 x - i x$

Step 2

Factor $y$ out of $2 i y + 5 y$ .

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Step 2.1

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

$y (2 i) + 5 y = 7 + 5 i - 3 x - i x$

Step 2.2

Factor $y$ out of $5 y$ .

$y (2 i) + y \cdot 5 = 7 + 5 i - 3 x - i x$

Step 2.3

Factor $y$ out of $y (2 i) + y \cdot 5$ .

$y (2 i + 5) = 7 + 5 i - 3 x - i x$

Step 3

Divide each term in $y (2 i + 5) = 7 + 5 i - 3 x - i x$ by $2 i + 5$ and simplify.

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Step 3.1

Divide each term in $y (2 i + 5) = 7 + 5 i - 3 x - i x$ by $2 i + 5$ .

$\frac{y (2 i + 5)}{2 i + 5} = \frac{7}{2 i + 5} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2 i + 5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{y (2 i + 5)}{2 i + 5} = \frac{7}{2 i + 5} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{7}{2 i + 5} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply the numerator and denominator of $\frac{7}{5 + 2 i}$ by the conjugate of $5 + 2 i$ to make the denominator real.

$y = \frac{7}{5 + 2 i} \cdot \frac{5 - 2 i}{5 - 2 i} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Combine.

$y = \frac{7 (5 - 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.2

Simplify the numerator.

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Step 3.3.1.2.2.1

Apply the distributive property.

$y = \frac{7 \cdot 5 + 7 (- 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.2.2

Multiply $7$ by $5$ .

$y = \frac{35 + 7 (- 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.2.3

Multiply $- 2$ by $7$ .

$y = \frac{35 - 14 i}{(5 + 2 i) (5 - 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3

Simplify the denominator.

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Step 3.3.1.2.3.1

Expand $(5 + 2 i) (5 - 2 i)$ using the FOIL Method.

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Step 3.3.1.2.3.1.1

Apply the distributive property.

$y = \frac{35 - 14 i}{5 (5 - 2 i) + 2 i (5 - 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.1.2

Apply the distributive property.

$y = \frac{35 - 14 i}{5 \cdot 5 + 5 (- 2 i) + 2 i (5 - 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.1.3

Apply the distributive property.

$y = \frac{35 - 14 i}{5 \cdot 5 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2

Simplify.

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Step 3.3.1.2.3.2.1

Multiply $5$ by $5$ .

$y = \frac{35 - 14 i}{25 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.2

Multiply $- 2$ by $5$ .

$y = \frac{35 - 14 i}{25 - 10 i + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.3

Multiply $5$ by $2$ .

$y = \frac{35 - 14 i}{25 - 10 i + 10 i + 2 i (- 2 i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.4

Multiply $- 2$ by $2$ .

$y = \frac{35 - 14 i}{25 - 10 i + 10 i - 4 i i} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.5

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

$y = \frac{35 - 14 i}{25 - 10 i + 10 i - 4 (i^{1} i)} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.6

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

$y = \frac{35 - 14 i}{25 - 10 i + 10 i - 4 (i^{1} i^{1})} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.7

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

$y = \frac{35 - 14 i}{25 - 10 i + 10 i - 4 i^{1 + 1}} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.8

Add $1$ and $1$ .

$y = \frac{35 - 14 i}{25 - 10 i + 10 i - 4 i^{2}} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.9

Add $- 10 i$ and $10 i$ .

$y = \frac{35 - 14 i}{25 + 0 - 4 i^{2}} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.2.10

Add $25$ and $0$ .

$y = \frac{35 - 14 i}{25 - 4 i^{2}} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.3

Simplify each term.

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Step 3.3.1.2.3.3.1

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

$y = \frac{35 - 14 i}{25 - 4 \cdot - 1} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{35 - 14 i}{25 + 4} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.2.3.4

Add $25$ and $4$ .

$y = \frac{35 - 14 i}{29} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.3

Split the fraction $\frac{35 - 14 i}{29}$ into two fractions.

$y = \frac{35}{29} + \frac{- 14 i}{29} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.4

Move the negative in front of the fraction.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{5 i}{2 i + 5} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.5

Multiply the numerator and denominator of $\frac{5 i}{5 + 2 i}$ by the conjugate of $5 + 2 i$ to make the denominator real.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{5 i}{5 + 2 i} \cdot \frac{5 - 2 i}{5 - 2 i} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6

Multiply.

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Step 3.3.1.6.1

Combine.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{5 i (5 - 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2

Simplify the numerator.

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Step 3.3.1.6.2.1

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{5 i \cdot 5 + 5 i (- 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.2

Multiply $5$ by $5$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{25 i + 5 i (- 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.3

Multiply $5 i (- 2 i)$ .

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Step 3.3.1.6.2.3.1

Multiply $- 2$ by $5$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{25 i - 10 i i}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.3.2

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{25 i - 10 (i^{1} i)}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.3.3

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{25 i - 10 (i^{1} i^{1})}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.3.4

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{25 i - 10 i^{1 + 1}}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.3.5

Add $1$ and $1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{25 i - 10 i^{2}}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.4

Simplify each term.

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Step 3.3.1.6.2.4.1

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{25 i - 10 \cdot - 1}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.4.2

Multiply $- 10$ by $- 1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{25 i + 10}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.2.5

Reorder $25 i$ and $10$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{(5 + 2 i) (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3

Simplify the denominator.

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Step 3.3.1.6.3.1

Expand $(5 + 2 i) (5 - 2 i)$ using the FOIL Method.

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Step 3.3.1.6.3.1.1

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{5 (5 - 2 i) + 2 i (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.1.2

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{5 \cdot 5 + 5 (- 2 i) + 2 i (5 - 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.1.3

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{5 \cdot 5 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2

Simplify.

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Step 3.3.1.6.3.2.1

Multiply $5$ by $5$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.2

Multiply $- 2$ by $5$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 10 i + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.3

Multiply $5$ by $2$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 10 i + 10 i + 2 i (- 2 i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.4

Multiply $- 2$ by $2$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 10 i + 10 i - 4 i i} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.5

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 10 i + 10 i - 4 (i^{1} i)} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.6

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 10 i + 10 i - 4 (i^{1} i^{1})} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.7

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 10 i + 10 i - 4 i^{1 + 1}} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.8

Add $1$ and $1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 10 i + 10 i - 4 i^{2}} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.9

Add $- 10 i$ and $10 i$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 + 0 - 4 i^{2}} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.2.10

Add $25$ and $0$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 4 i^{2}} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.3

Simplify each term.

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Step 3.3.1.6.3.3.1

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 - 4 \cdot - 1} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{25 + 4} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.6.3.4

Add $25$ and $4$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10 + 25 i}{29} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.7

Split the fraction $\frac{10 + 25 i}{29}$ into two fractions.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 3 x}{2 i + 5} + \frac{- i x}{2 i + 5}$

Step 3.3.1.8

Multiply the numerator and denominator of $\frac{- 3 x}{5 + 2 i}$ by the conjugate of $5 + 2 i$ to make the denominator real.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 3 x}{5 + 2 i} \cdot \frac{5 - 2 i}{5 - 2 i} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9

Multiply.

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Step 3.3.1.9.1

Combine.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 3 x (5 - 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.2

Simplify the numerator.

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Step 3.3.1.9.2.1

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 3 x \cdot 5 - 3 x (- 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.2.2

Multiply $5$ by $- 3$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x - 3 x (- 2 i)}{(5 + 2 i) (5 - 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.2.3

Multiply $- 2$ by $- 3$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{(5 + 2 i) (5 - 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3

Simplify the denominator.

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Step 3.3.1.9.3.1

Expand $(5 + 2 i) (5 - 2 i)$ using the FOIL Method.

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Step 3.3.1.9.3.1.1

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{5 (5 - 2 i) + 2 i (5 - 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.1.2

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{5 \cdot 5 + 5 (- 2 i) + 2 i (5 - 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.1.3

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{5 \cdot 5 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2

Simplify.

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Step 3.3.1.9.3.2.1

Multiply $5$ by $5$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.2

Multiply $- 2$ by $5$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 10 i + 2 i \cdot 5 + 2 i (- 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.3

Multiply $5$ by $2$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 10 i + 10 i + 2 i (- 2 i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.4

Multiply $- 2$ by $2$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 10 i + 10 i - 4 i i} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.5

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 10 i + 10 i - 4 (i^{1} i)} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.6

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 10 i + 10 i - 4 (i^{1} i^{1})} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.7

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 10 i + 10 i - 4 i^{1 + 1}} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.8

Add $1$ and $1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 10 i + 10 i - 4 i^{2}} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.9

Add $- 10 i$ and $10 i$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 + 0 - 4 i^{2}} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.2.10

Add $25$ and $0$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 4 i^{2}} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.3

Simplify each term.

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Step 3.3.1.9.3.3.1

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 - 4 \cdot - 1} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{25 + 4} + \frac{- i x}{2 i + 5}$

Step 3.3.1.9.3.4

Add $25$ and $4$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{- 15 x + 6 x i}{29} + \frac{- i x}{2 i + 5}$

Step 3.3.1.10

Factor $3 x$ out of $- 15 x + 6 x i$ .

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Step 3.3.1.10.1

Factor $3 x$ out of $- 15 x$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5) + 6 x i}{29} + \frac{- i x}{2 i + 5}$

Step 3.3.1.10.2

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5) + 3 x (2 i)}{29} + \frac{- i x}{2 i + 5}$

Step 3.3.1.10.3

Factor $3 x$ out of $3 x (- 5) + 3 x (2 i)$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- i x}{2 i + 5}$

Step 3.3.1.11

Multiply the numerator and denominator of $\frac{- i x}{5 + 2 i}$ by the conjugate of $5 + 2 i$ to make the denominator real.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- i x}{5 + 2 i} \cdot \frac{5 - 2 i}{5 - 2 i}$

Step 3.3.1.12

Multiply.

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Step 3.3.1.12.1

Combine.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- i x (5 - 2 i)}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2

Simplify the numerator.

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Step 3.3.1.12.2.1

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- i x \cdot 5 - i x (- 2 i)}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2.2

Multiply $5$ by $- 1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - i x (- 2 i)}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2.3

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

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Step 3.3.1.12.2.3.1

Multiply $- 2$ by $- 1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x + 2 i x i}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2.3.2

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x + 2 x (i^{1} i)}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2.3.3

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x + 2 x (i^{1} i^{1})}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2.3.4

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x + 2 x i^{1 + 1}}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2.3.5

Add $1$ and $1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x + 2 x i^{2}}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2.4

Simplify each term.

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Step 3.3.1.12.2.4.1

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x + 2 x \cdot - 1}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.2.4.2

Multiply $- 1$ by $2$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{(5 + 2 i) (5 - 2 i)}$

Step 3.3.1.12.3

Simplify the denominator.

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Step 3.3.1.12.3.1

Expand $(5 + 2 i) (5 - 2 i)$ using the FOIL Method.

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Step 3.3.1.12.3.1.1

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{5 (5 - 2 i) + 2 i (5 - 2 i)}$

Step 3.3.1.12.3.1.2

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{5 \cdot 5 + 5 (- 2 i) + 2 i (5 - 2 i)}$

Step 3.3.1.12.3.1.3

Apply the distributive property.

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{5 \cdot 5 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)}$

Step 3.3.1.12.3.2

Simplify.

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Step 3.3.1.12.3.2.1

Multiply $5$ by $5$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)}$

Step 3.3.1.12.3.2.2

Multiply $- 2$ by $5$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 10 i + 2 i \cdot 5 + 2 i (- 2 i)}$

Step 3.3.1.12.3.2.3

Multiply $5$ by $2$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 10 i + 10 i + 2 i (- 2 i)}$

Step 3.3.1.12.3.2.4

Multiply $- 2$ by $2$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 10 i + 10 i - 4 i i}$

Step 3.3.1.12.3.2.5

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 10 i + 10 i - 4 (i^{1} i)}$

Step 3.3.1.12.3.2.6

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 10 i + 10 i - 4 (i^{1} i^{1})}$

Step 3.3.1.12.3.2.7

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 10 i + 10 i - 4 i^{1 + 1}}$

Step 3.3.1.12.3.2.8

Add $1$ and $1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 10 i + 10 i - 4 i^{2}}$

Step 3.3.1.12.3.2.9

Add $- 10 i$ and $10 i$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 + 0 - 4 i^{2}}$

Step 3.3.1.12.3.2.10

Add $25$ and $0$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 4 i^{2}}$

Step 3.3.1.12.3.3

Simplify each term.

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Step 3.3.1.12.3.3.1

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 - 4 \cdot - 1}$

Step 3.3.1.12.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{25 + 4}$

Step 3.3.1.12.3.4

Add $25$ and $4$ .

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{- 5 i x - 2 x}{29}$

Step 3.3.1.13

Factor $x$ out of $- 5 i x - 2 x$ .

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Step 3.3.1.13.1

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{x (- 5 i) - 2 x}{29}$

Step 3.3.1.13.2

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{x (- 5 i) + x \cdot - 2}{29}$

Step 3.3.1.13.3

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

$y = \frac{35}{29} - \frac{14 i}{29} + \frac{10}{29} + \frac{25 i}{29} + \frac{3 x (- 5 + 2 i)}{29} + \frac{x (- 5 i - 2)}{29}$

Step 3.3.2

Combine the numerators over the common denominator.

$y = \frac{35 - 14 i + 10 + 25 i + 3 x (- 5 + 2 i) + x (- 5 i - 2)}{29}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

Apply the distributive property.

$y = \frac{35 - 14 i + 10 + 25 i + 3 x \cdot - 5 + 3 x (2 i) + x (- 5 i - 2)}{29}$

Step 3.3.3.2

Multiply $- 5$ by $3$ .

$y = \frac{35 - 14 i + 10 + 25 i - 15 x + 3 x (2 i) + x (- 5 i - 2)}{29}$

Step 3.3.3.3

Multiply $2$ by $3$ .

$y = \frac{35 - 14 i + 10 + 25 i - 15 x + 6 x i + x (- 5 i - 2)}{29}$

Step 3.3.3.4

Apply the distributive property.

$y = \frac{35 - 14 i + 10 + 25 i - 15 x + 6 x i + x (- 5 i) + x \cdot - 2}{29}$

Step 3.3.3.5

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

$y = \frac{35 - 14 i + 10 + 25 i - 15 x + 6 x i + x (- 5 i) - 2 x}{29}$

Step 3.3.4

Simplify by adding terms.

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Step 3.3.4.1

Add $35$ and $10$ .

$y = \frac{45 - 14 i + 25 i - 15 x + 6 x i + x (- 5 i) - 2 x}{29}$

Step 3.3.4.2

Add $- 14 i$ and $25 i$ .

$y = \frac{45 + 11 i - 15 x + 6 x i + x (- 5 i) - 2 x}{29}$

Step 3.3.4.3

Subtract $2 x$ from $- 15 x$ .

$y = \frac{45 + 11 i + 6 x i + x (- 5 i) - 17 x}{29}$

Step 3.3.5

Add $6 x i$ and $x (- 5 i)$ .

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Step 3.3.5.1

Reorder $x$ and $- 5$ .

$y = \frac{45 + 11 i + 6 x i - 5 \cdot x i - 17 x}{29}$

Step 3.3.5.2

Subtract $5 \cdot x i$ from $6 x i$ .

$y = \frac{45 + 11 i + 1 x i - 17 x}{29}$

Step 3.3.6

Multiply $x$ by $1$ .

$y = \frac{45 + 11 i + x i - 17 x}{29}$

Step 3.3.7

Reorder terms.

$y = \frac{i x - 17 x + 45 + 11 i}{29}$