Precalculus Examples

$(2 x - y) - 16 i = 10 + 4 y i$

Step 1

Subtract $4 y i$ from both sides of the equation.

$2 x - y - 16 i - 4 y i = 10$

Step 2

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

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Step 2.1

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

$- y - 16 i - 4 y i = 10 - 2 x$

Step 2.2

Add $16 i$ to both sides of the equation.

$- y - 4 y i = 10 - 2 x + 16 i$

Step 3

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

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Step 3.1

Factor $y$ out of $- y$ .

$y \cdot - 1 - 4 y i = 10 - 2 x + 16 i$

Step 3.2

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

$y \cdot - 1 + y (- 4 i) = 10 - 2 x + 16 i$

Step 3.3

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

$y (- 1 - 4 i) = 10 - 2 x + 16 i$

Step 4

Divide each term in $y (- 1 - 4 i) = 10 - 2 x + 16 i$ by $- 1 - 4 i$ and simplify.

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Step 4.1

Divide each term in $y (- 1 - 4 i) = 10 - 2 x + 16 i$ by $- 1 - 4 i$ .

$\frac{y (- 1 - 4 i)}{- 1 - 4 i} = \frac{10}{- 1 - 4 i} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 1 - 4 i$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{y (- 1 - 4 i)}{- 1 - 4 i} = \frac{10}{- 1 - 4 i} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{10}{- 1 - 4 i} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify terms.

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Step 4.3.1.1

Simplify each term.

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Step 4.3.1.1.1

Multiply the numerator and denominator of $\frac{10}{- 1 - 4 i}$ by the conjugate of $- 1 - 4 i$ to make the denominator real.

$y = \frac{10}{- 1 - 4 i} \cdot \frac{- 1 + 4 i}{- 1 + 4 i} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2

Multiply.

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Step 4.3.1.1.2.1

Combine.

$y = \frac{10 (- 1 + 4 i)}{(- 1 - 4 i) (- 1 + 4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.2

Simplify the numerator.

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Step 4.3.1.1.2.2.1

Apply the distributive property.

$y = \frac{10 \cdot - 1 + 10 (4 i)}{(- 1 - 4 i) (- 1 + 4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.2.2

Multiply $10$ by $- 1$ .

$y = \frac{- 10 + 10 (4 i)}{(- 1 - 4 i) (- 1 + 4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.2.3

Multiply $4$ by $10$ .

$y = \frac{- 10 + 40 i}{(- 1 - 4 i) (- 1 + 4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3

Simplify the denominator.

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Step 4.3.1.1.2.3.1

Expand $(- 1 - 4 i) (- 1 + 4 i)$ using the FOIL Method.

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Step 4.3.1.1.2.3.1.1

Apply the distributive property.

$y = \frac{- 10 + 40 i}{- 1 (- 1 + 4 i) - 4 i (- 1 + 4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.1.2

Apply the distributive property.

$y = \frac{- 10 + 40 i}{- 1 \cdot - 1 - 1 (4 i) - 4 i (- 1 + 4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.1.3

Apply the distributive property.

$y = \frac{- 10 + 40 i}{- 1 \cdot - 1 - 1 (4 i) - 4 i \cdot - 1 - 4 i (4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2

Simplify.

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Step 4.3.1.1.2.3.2.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 10 + 40 i}{1 - 1 (4 i) - 4 i \cdot - 1 - 4 i (4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.2

Multiply $4$ by $- 1$ .

$y = \frac{- 10 + 40 i}{1 - 4 i - 4 i \cdot - 1 - 4 i (4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.3

Multiply $- 1$ by $- 4$ .

$y = \frac{- 10 + 40 i}{1 - 4 i + 4 i - 4 i (4 i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.4

Multiply $4$ by $- 4$ .

$y = \frac{- 10 + 40 i}{1 - 4 i + 4 i - 16 i i} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.5

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

$y = \frac{- 10 + 40 i}{1 - 4 i + 4 i - 16 (i^{1} i)} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.6

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

$y = \frac{- 10 + 40 i}{1 - 4 i + 4 i - 16 (i^{1} i^{1})} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.7

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

$y = \frac{- 10 + 40 i}{1 - 4 i + 4 i - 16 i^{1 + 1}} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.8

Add $1$ and $1$ .

$y = \frac{- 10 + 40 i}{1 - 4 i + 4 i - 16 i^{2}} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.9

Add $- 4 i$ and $4 i$ .

$y = \frac{- 10 + 40 i}{1 + 0 - 16 i^{2}} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.2.10

Add $1$ and $0$ .

$y = \frac{- 10 + 40 i}{1 - 16 i^{2}} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.3

Simplify each term.

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Step 4.3.1.1.2.3.3.1

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

$y = \frac{- 10 + 40 i}{1 - 16 \cdot - 1} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.3.2

Multiply $- 16$ by $- 1$ .

$y = \frac{- 10 + 40 i}{1 + 16} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.2.3.4

Add $1$ and $16$ .

$y = \frac{- 10 + 40 i}{17} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.3

Split the fraction $\frac{- 10 + 40 i}{17}$ into two fractions.

$y = \frac{- 10}{17} + \frac{40 i}{17} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.4

Move the negative in front of the fraction.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{- 2 x}{- 1 - 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.5

Multiply the numerator and denominator of $\frac{- 2 x}{- 1 - 4 i}$ by the conjugate of $- 1 - 4 i$ to make the denominator real.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{- 2 x}{- 1 - 4 i} \cdot \frac{- 1 + 4 i}{- 1 + 4 i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6

Multiply.

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Step 4.3.1.1.6.1

Combine.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{- 2 x (- 1 + 4 i)}{(- 1 - 4 i) (- 1 + 4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.2

Simplify the numerator.

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Step 4.3.1.1.6.2.1

Apply the distributive property.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{- 2 x \cdot - 1 - 2 x (4 i)}{(- 1 - 4 i) (- 1 + 4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.2.2

Multiply $- 1$ by $- 2$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 2 x (4 i)}{(- 1 - 4 i) (- 1 + 4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.2.3

Multiply $4$ by $- 2$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{(- 1 - 4 i) (- 1 + 4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3

Simplify the denominator.

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Step 4.3.1.1.6.3.1

Expand $(- 1 - 4 i) (- 1 + 4 i)$ using the FOIL Method.

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Step 4.3.1.1.6.3.1.1

Apply the distributive property.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{- 1 (- 1 + 4 i) - 4 i (- 1 + 4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.1.2

Apply the distributive property.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{- 1 \cdot - 1 - 1 (4 i) - 4 i (- 1 + 4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.1.3

Apply the distributive property.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{- 1 \cdot - 1 - 1 (4 i) - 4 i \cdot - 1 - 4 i (4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2

Simplify.

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Step 4.3.1.1.6.3.2.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 1 (4 i) - 4 i \cdot - 1 - 4 i (4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.2

Multiply $4$ by $- 1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 4 i - 4 i \cdot - 1 - 4 i (4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.3

Multiply $- 1$ by $- 4$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 4 i + 4 i - 4 i (4 i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.4

Multiply $4$ by $- 4$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 4 i + 4 i - 16 i i} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.5

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 4 i + 4 i - 16 (i^{1} i)} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.6

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 4 i + 4 i - 16 (i^{1} i^{1})} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.7

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 4 i + 4 i - 16 i^{1 + 1}} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.8

Add $1$ and $1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 4 i + 4 i - 16 i^{2}} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.9

Add $- 4 i$ and $4 i$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 + 0 - 16 i^{2}} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.2.10

Add $1$ and $0$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 16 i^{2}} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.3

Simplify each term.

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Step 4.3.1.1.6.3.3.1

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 - 16 \cdot - 1} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.3.2

Multiply $- 16$ by $- 1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{1 + 16} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.6.3.4

Add $1$ and $16$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x - 8 x i}{17} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.7

Factor $2 x$ out of $2 x - 8 x i$ .

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Step 4.3.1.1.7.1

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1) - 8 x i}{17} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.7.2

Factor $2 x$ out of $- 8 x i$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1) + 2 x (- 4 i)}{17} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.7.3

Factor $2 x$ out of $2 x (1) + 2 x (- 4 i)$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{16 i}{- 1 - 4 i}$

Step 4.3.1.1.8

Multiply the numerator and denominator of $\frac{16 i}{- 1 - 4 i}$ by the conjugate of $- 1 - 4 i$ to make the denominator real.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{16 i}{- 1 - 4 i} \cdot \frac{- 1 + 4 i}{- 1 + 4 i}$

Step 4.3.1.1.9

Multiply.

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Step 4.3.1.1.9.1

Combine.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{16 i (- 1 + 4 i)}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2

Simplify the numerator.

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Step 4.3.1.1.9.2.1

Apply the distributive property.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{16 i \cdot - 1 + 16 i (4 i)}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.2

Multiply $- 1$ by $16$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 16 i + 16 i (4 i)}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.3

Multiply $16 i (4 i)$ .

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Step 4.3.1.1.9.2.3.1

Multiply $4$ by $16$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 16 i + 64 i i}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.3.2

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 16 i + 64 (i^{1} i)}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.3.3

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 16 i + 64 (i^{1} i^{1})}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.3.4

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 16 i + 64 i^{1 + 1}}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.3.5

Add $1$ and $1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 16 i + 64 i^{2}}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.4

Simplify each term.

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Step 4.3.1.1.9.2.4.1

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 16 i + 64 \cdot - 1}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.4.2

Multiply $64$ by $- 1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 16 i - 64}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.2.5

Reorder $- 16 i$ and $- 64$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{(- 1 - 4 i) (- 1 + 4 i)}$

Step 4.3.1.1.9.3

Simplify the denominator.

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Step 4.3.1.1.9.3.1

Expand $(- 1 - 4 i) (- 1 + 4 i)$ using the FOIL Method.

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Step 4.3.1.1.9.3.1.1

Apply the distributive property.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{- 1 (- 1 + 4 i) - 4 i (- 1 + 4 i)}$

Step 4.3.1.1.9.3.1.2

Apply the distributive property.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{- 1 \cdot - 1 - 1 (4 i) - 4 i (- 1 + 4 i)}$

Step 4.3.1.1.9.3.1.3

Apply the distributive property.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{- 1 \cdot - 1 - 1 (4 i) - 4 i \cdot - 1 - 4 i (4 i)}$

Step 4.3.1.1.9.3.2

Simplify.

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Step 4.3.1.1.9.3.2.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 1 (4 i) - 4 i \cdot - 1 - 4 i (4 i)}$

Step 4.3.1.1.9.3.2.2

Multiply $4$ by $- 1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 4 i - 4 i \cdot - 1 - 4 i (4 i)}$

Step 4.3.1.1.9.3.2.3

Multiply $- 1$ by $- 4$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 4 i + 4 i - 4 i (4 i)}$

Step 4.3.1.1.9.3.2.4

Multiply $4$ by $- 4$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 4 i + 4 i - 16 i i}$

Step 4.3.1.1.9.3.2.5

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 4 i + 4 i - 16 (i^{1} i)}$

Step 4.3.1.1.9.3.2.6

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 4 i + 4 i - 16 (i^{1} i^{1})}$

Step 4.3.1.1.9.3.2.7

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 4 i + 4 i - 16 i^{1 + 1}}$

Step 4.3.1.1.9.3.2.8

Add $1$ and $1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 4 i + 4 i - 16 i^{2}}$

Step 4.3.1.1.9.3.2.9

Add $- 4 i$ and $4 i$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 + 0 - 16 i^{2}}$

Step 4.3.1.1.9.3.2.10

Add $1$ and $0$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 16 i^{2}}$

Step 4.3.1.1.9.3.3

Simplify each term.

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Step 4.3.1.1.9.3.3.1

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

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 - 16 \cdot - 1}$

Step 4.3.1.1.9.3.3.2

Multiply $- 16$ by $- 1$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{1 + 16}$

Step 4.3.1.1.9.3.4

Add $1$ and $16$ .

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64 - 16 i}{17}$

Step 4.3.1.1.10

Split the fraction $\frac{- 64 - 16 i}{17}$ into two fractions.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} + \frac{- 64}{17} + \frac{- 16 i}{17}$

Step 4.3.1.1.11

Simplify each term.

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Step 4.3.1.1.11.1

Move the negative in front of the fraction.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} - \frac{64}{17} + \frac{- 16 i}{17}$

Step 4.3.1.1.11.2

Move the negative in front of the fraction.

$y = - \frac{10}{17} + \frac{40 i}{17} + \frac{2 x (1 - 4 i)}{17} - \frac{64}{17} - \frac{16 i}{17}$

Step 4.3.1.2

Combine the numerators over the common denominator.

$y = \frac{- 10 + 40 i + 2 x (1 - 4 i) - 64 - 16 i}{17}$

Step 4.3.1.3

Simplify each term.

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Step 4.3.1.3.1

Apply the distributive property.

$y = \frac{- 10 + 40 i + 2 x \cdot 1 + 2 x (- 4 i) - 64 - 16 i}{17}$

Step 4.3.1.3.2

Multiply $2$ by $1$ .

$y = \frac{- 10 + 40 i + 2 x + 2 x (- 4 i) - 64 - 16 i}{17}$

Step 4.3.1.3.3

Multiply $- 4$ by $2$ .

$y = \frac{- 10 + 40 i + 2 x - 8 x i - 64 - 16 i}{17}$

Step 4.3.1.4

Simplify by adding terms.

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Step 4.3.1.4.1

Subtract $64$ from $- 10$ .

$y = \frac{40 i + 2 x - 8 x i - 74 - 16 i}{17}$

Step 4.3.1.4.2

Subtract $16 i$ from $40 i$ .

$y = \frac{2 x - 8 x i - 74 + 24 i}{17}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $2$ out of $2 x - 8 x i - 74 + 24 i$ .

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Step 4.3.2.1.1

Factor $2$ out of $2 x$ .

$y = \frac{2 (x) - 8 x i - 74 + 24 i}{17}$

Step 4.3.2.1.2

Factor $2$ out of $- 8 x i$ .

$y = \frac{2 (x) + 2 (- 4 x i) - 74 + 24 i}{17}$

Step 4.3.2.1.3

Factor $2$ out of $- 74$ .

$y = \frac{2 x + 2 (- 4 x i) + 2 \cdot - 37 + 24 i}{17}$

Step 4.3.2.1.4

Factor $2$ out of $24 i$ .

$y = \frac{2 x + 2 (- 4 x i) + 2 \cdot - 37 + 2 (12 i)}{17}$

Step 4.3.2.1.5

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

$y = \frac{2 (x - 4 x i) + 2 \cdot - 37 + 2 (12 i)}{17}$

Step 4.3.2.1.6

Factor $2$ out of $2 (x - 4 x i) + 2 \cdot - 37$ .

$y = \frac{2 (x - 4 x i - 37) + 2 (12 i)}{17}$

Step 4.3.2.1.7

Factor $2$ out of $2 (x - 4 x i - 37) + 2 (12 i)$ .

$y = \frac{2 (x - 4 x i - 37 + 12 i)}{17}$

Step 4.3.2.2

Reorder terms.

$y = \frac{2 (- 4 i x + x - 37 + 12 i)}{17}$

Step 4.3.3

Simplify with factoring out.

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Step 4.3.3.1

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

$y = \frac{2 (- (4 i x) + x - 37 + 12 i)}{17}$

Step 4.3.3.2

Factor $- 1$ out of $x$ .

$y = \frac{2 (- (4 i x) - 1 (- x) - 37 + 12 i)}{17}$

Step 4.3.3.3

Factor $- 1$ out of $- (4 i x) - 1 (- x)$ .

$y = \frac{2 (- (4 i x - x) - 37 + 12 i)}{17}$

Step 4.3.3.4

Rewrite $- 37$ as $- 1 (37)$ .

$y = \frac{2 (- (4 i x - x) - 1 (37) + 12 i)}{17}$

Step 4.3.3.5

Factor $- 1$ out of $- (4 i x - x) - 1 (37)$ .

$y = \frac{2 (- (4 i x - x + 37) + 12 i)}{17}$

Step 4.3.3.6

Factor $- 1$ out of $12 i$ .

$y = \frac{2 (- (4 i x - x + 37) - (- 12 i))}{17}$

Step 4.3.3.7

Factor $- 1$ out of $- (4 i x - x + 37) - (- 12 i)$ .

$y = \frac{2 (- (4 i x - x + 37 - 12 i))}{17}$

Step 4.3.3.8

Simplify the expression.

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Step 4.3.3.8.1

Rewrite $- (4 i x - x + 37 - 12 i)$ as $- 1 (4 i x - x + 37 - 12 i)$ .

$y = \frac{2 (- 1 (4 i x - x + 37 - 12 i))}{17}$

Step 4.3.3.8.2

Move the negative in front of the fraction.

$y = - \frac{2 (4 i x - x + 37 - 12 i)}{17}$