Finite Math Examples

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - (i + 1) \\ 4 & {(i - 1)}^{2} & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 1

Apply the distributive property.

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \cdot 1 \\ 4 & {(i - 1)}^{2} & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 2

Multiply $- 1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & {(i - 1)}^{2} & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 3

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & (i - 1) (i - 1) & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 4

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

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Step 4.1

Apply the distributive property.

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & i (i - 1) - 1 (i - 1) & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 4.2

Apply the distributive property.

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & i i + i \cdot - 1 - 1 (i - 1) & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 4.3

Apply the distributive property.

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & i i + i \cdot - 1 - 1 i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $i i$ .

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Step 5.1.1.1

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & i^{1} i + i \cdot - 1 - 1 i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.1.1.2

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & i^{1} i^{1} + i \cdot - 1 - 1 i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.1.1.3

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & i^{1 + 1} + i \cdot - 1 - 1 i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.1.1.4

Add $1$ and $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & i^{2} + i \cdot - 1 - 1 i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.1.2

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 1 + i \cdot - 1 - 1 i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.1.3

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 1 - 1 \cdot i - 1 i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.1.4

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 1 - i - 1 i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.1.5

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 1 - i - i - 1 \cdot - 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.1.6

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 1 - i - i + 1 & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.2

Add $- 1$ and $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & 0 - i - i & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.3

Subtract $i$ from $0$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - i - i & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 5.4

Subtract $i$ from $- i$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & {(i + 1)}^{2} \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 6

Rewrite ${(i + 1)}^{2}$ as $(i + 1) (i + 1)$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & (i + 1) (i + 1) \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 7

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

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Step 7.1

Apply the distributive property.

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & i (i + 1) + 1 (i + 1) \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 7.2

Apply the distributive property.

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & i i + i \cdot 1 + 1 (i + 1) \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 7.3

Apply the distributive property.

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & i i + i \cdot 1 + 1 i + 1 \cdot 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $i i$ .

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Step 8.1.1.1

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & i^{1} i + i \cdot 1 + 1 i + 1 \cdot 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.1.1.2

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & i^{1} i^{1} + i \cdot 1 + 1 i + 1 \cdot 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.1.1.3

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & i^{1 + 1} + i \cdot 1 + 1 i + 1 \cdot 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.1.1.4

Add $1$ and $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & i^{2} + i \cdot 1 + 1 i + 1 \cdot 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.1.2

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

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & - 1 + i \cdot 1 + 1 i + 1 \cdot 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.1.3

Multiply $i$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & - 1 + i + 1 i + 1 \cdot 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.1.4

Multiply $i$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & - 1 + i + i + 1 \cdot 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.1.5

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & - 1 + i + i + 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.2

Add $- 1$ and $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & 0 + i + i \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.3

Add $0$ and $i$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & i + i \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 8.4

Add $i$ and $i$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & 2 i \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 9

Multiply $[\begin{array}{c} 1 & 1 & 1 \\ - 2 & i - 1 & - i - 1 \\ 4 & - 2 i & 2 i \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

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Step 9.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $3 \times 3$ and the second matrix is $3 \times 1$ .

Step 9.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 1 x + 1 y + 1 z \\ - 2 x + (i - 1) y + (- i - 1) z \\ 4 x - 2 i y + 2 i z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 9.3

Simplify each element of the matrix by multiplying out all the expressions.

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Step 9.3.1

Apply the distributive property.

$[\begin{array}{c} x + y + z \\ - 2 x + i y - 1 y + (- i - 1) z \\ 4 x - 2 i y + 2 i z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 9.3.2

Rewrite $- 1 y$ as $- y$ .

$[\begin{array}{c} x + y + z \\ - 2 x + i y - y + (- i - 1) z \\ 4 x - 2 i y + 2 i z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 9.3.3

Apply the distributive property.

$[\begin{array}{c} x + y + z \\ - 2 x + i y - y - i z - 1 z \\ 4 x - 2 i y + 2 i z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 9.3.4

Rewrite $- 1 z$ as $- z$ .

$[\begin{array}{c} x + y + z \\ - 2 x + i y - y - i z - z \\ 4 x - 2 i y + 2 i z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}]$

Step 10

Write as a linear system of equations.

$x + y + z = 0$

$- 2 x + i y - y - i z - z = 0$

$4 x - 2 i y + 2 i z = 1$

Step 11

Solve the system of equations.

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Step 11.1

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

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Step 11.1.1

Subtract $y$ from both sides of the equation.

$x + z = - y$

$- 2 x + i y - y - i z - z = 0$

$4 x - 2 i y + 2 i z = 1$

Step 11.1.2

Subtract $z$ from both sides of the equation.

$x = - y - z$

$- 2 x + i y - y - i z - z = 0$

$4 x - 2 i y + 2 i z = 1$

$x = - y - z$

$- 2 x + i y - y - i z - z = 0$

$4 x - 2 i y + 2 i z = 1$

Step 11.2

Replace all occurrences of $x$ with $- y - z$ in each equation.

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Step 11.2.1

Replace all occurrences of $x$ in $- 2 x + i y - y - i z - z = 0$ with $- y - z$ .

$- 2 (- y - z) + i y - y - i z - z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

Simplify $- 2 (- y - z) + i y - y - i z - z$ .

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Step 11.2.2.1.1

Simplify each term.

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Step 11.2.2.1.1.1

Apply the distributive property.

$- 2 (- y) - 2 (- z) + i y - y - i z - z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

Step 11.2.2.1.1.2

Multiply $- 1$ by $- 2$ .

$2 y - 2 (- z) + i y - y - i z - z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

Step 11.2.2.1.1.3

Multiply $- 1$ by $- 2$ .

$2 y + 2 z + i y - y - i z - z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

$2 y + 2 z + i y - y - i z - z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

Step 11.2.2.1.2

Simplify by adding terms.

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Step 11.2.2.1.2.1

Subtract $y$ from $2 y$ .

$y + 2 z + i y - i z - z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

Step 11.2.2.1.2.2

Subtract $z$ from $2 z$ .

$y + i y - i z + z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

$4 x - 2 i y + 2 i z = 1$

Step 11.2.3

Replace all occurrences of $x$ in $4 x - 2 i y + 2 i z = 1$ with $- y - z$ .

$4 (- y - z) - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.2.4

Simplify the left side.

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Step 11.2.4.1

Simplify each term.

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Step 11.2.4.1.1

Apply the distributive property.

$4 (- y) + 4 (- z) - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.2.4.1.2

Multiply $- 1$ by $4$ .

$- 4 y + 4 (- z) - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.2.4.1.3

Multiply $- 1$ by $4$ .

$- 4 y - 4 z - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

$- 4 y - 4 z - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

$- 4 y - 4 z - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

$- 4 y - 4 z - 2 i y + 2 i z = 1$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3

Solve for $y$ in $- 4 y - 4 z - 2 i y + 2 i z = 1$ .

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Step 11.3.1

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

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Step 11.3.1.1

Add $4 z$ to both sides of the equation.

$- 4 y - 2 i y + 2 i z = 1 + 4 z$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.1.2

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

$- 4 y - 2 i y = 1 + 4 z - 2 i z$

$y + i y - i z + z = 0$

$x = - y - z$

$- 4 y - 2 i y = 1 + 4 z - 2 i z$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.2

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

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Step 11.3.2.1

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

$2 y (- 2) - 2 i y = 1 + 4 z - 2 i z$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.2.2

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

$2 y (- 2) + 2 y (- i) = 1 + 4 z - 2 i z$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.2.3

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

$2 y (- 2 - i) = 1 + 4 z - 2 i z$

$y + i y - i z + z = 0$

$x = - y - z$

$2 y (- 2 - i) = 1 + 4 z - 2 i z$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3

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

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Step 11.3.3.1

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.2

Simplify the left side.

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Step 11.3.3.2.1

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

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Step 11.3.3.2.1.1

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.2.1.2

Cancel the common factors.

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Step 11.3.3.2.1.2.1

Factor $2$ out of $- 4$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.2.1.2.2

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.2.1.2.3

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.2.1.2.4

Cancel the common factor.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.2.1.2.5

Rewrite the expression.

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.2.2

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

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Step 11.3.3.2.2.1

Cancel the common factor.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.2.2.2

Divide $y$ by $1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3

Simplify the right side.

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Step 11.3.3.3.1

Simplify each term.

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Step 11.3.3.3.1.1

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2

Multiply.

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Step 11.3.3.3.1.2.1

Combine.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.2

Multiply $- 4 + 2 i$ by $1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3

Simplify the denominator.

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Step 11.3.3.3.1.2.3.1

Expand $(- 4 - 2 i) (- 4 + 2 i)$ using the FOIL Method.

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Step 11.3.3.3.1.2.3.1.1

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.1.2

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.1.3

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2

Simplify.

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Step 11.3.3.3.1.2.3.2.1

Multiply $- 4$ by $- 4$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.2

Multiply $2$ by $- 4$ .

$y = \frac{- 4 + 2 i}{16 - 8 i - 2 i \cdot - 4 - 2 i (2 i)} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.3

Multiply $- 4$ by $- 2$ .

$y = \frac{- 4 + 2 i}{16 - 8 i + 8 i - 2 i (2 i)} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.4

Multiply $2$ by $- 2$ .

$y = \frac{- 4 + 2 i}{16 - 8 i + 8 i - 4 i i} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.5

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

$y = \frac{- 4 + 2 i}{16 - 8 i + 8 i - 4 (i i)} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.6

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

$y = \frac{- 4 + 2 i}{16 - 8 i + 8 i - 4 (i i)} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.7

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

$y = \frac{- 4 + 2 i}{16 - 8 i + 8 i - 4 i^{1 + 1}} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.8

Add $1$ and $1$ .

$y = \frac{- 4 + 2 i}{16 - 8 i + 8 i - 4 i^{2}} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.9

Add $- 8 i$ and $8 i$ .

$y = \frac{- 4 + 2 i}{16 + 0 - 4 i^{2}} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.2.10

Add $16$ and $0$ .

$y = \frac{- 4 + 2 i}{16 - 4 i^{2}} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 4 + 2 i}{16 - 4 i^{2}} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.3

Simplify each term.

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Step 11.3.3.3.1.2.3.3.1

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.3.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 4 + 2 i}{16 + 4} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 4 + 2 i}{16 + 4} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.2.3.4

Add $16$ and $4$ .

$y = \frac{- 4 + 2 i}{20} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 4 + 2 i}{20} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 4 + 2 i}{20} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.3

Cancel the common factor of $- 4 + 2 i$ and $20$ .

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Step 11.3.3.3.1.3.1

Factor $2$ out of $- 4$ .

$y = \frac{2 \cdot - 2 + 2 i}{20} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.3.2

Factor $2$ out of $2 i$ .

$y = \frac{2 \cdot - 2 + 2 (i)}{20} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.3.3

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

$y = \frac{2 \cdot (- 2 + i)}{20} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.3.4

Cancel the common factors.

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Step 11.3.3.3.1.3.4.1

Factor $2$ out of $20$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.3.4.2

Cancel the common factor.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.3.4.3

Rewrite the expression.

$y = \frac{- 2 + i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 2 + i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 2 + i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.4

Split the fraction $\frac{- 2 + i}{10}$ into two fractions.

$y = \frac{- 2}{10} + \frac{i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.5

Simplify each term.

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Step 11.3.3.3.1.5.1

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

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Step 11.3.3.3.1.5.1.1

Factor $2$ out of $- 2$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.5.1.2

Cancel the common factors.

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Step 11.3.3.3.1.5.1.2.1

Factor $2$ out of $10$ .

$y = \frac{2 \cdot - 1}{2 \cdot 5} + \frac{i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.5.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 1}{2 \cdot 5} + \frac{i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.5.1.2.3

Rewrite the expression.

$y = \frac{- 1}{5} + \frac{i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 1}{5} + \frac{i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 1}{5} + \frac{i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.5.2

Move the negative in front of the fraction.

$y = - \frac{1}{5} + \frac{i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = - \frac{1}{5} + \frac{i}{10} + \frac{4 z}{- 4 - 2 i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.6

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

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Step 11.3.3.3.1.6.1

Factor $2$ out of $4 z$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.6.2

Cancel the common factors.

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Step 11.3.3.3.1.6.2.1

Factor $2$ out of $- 4$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.6.2.2

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.6.2.3

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.6.2.4

Cancel the common factor.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.6.2.5

Rewrite the expression.

$y = - \frac{1}{5} + \frac{i}{10} + \frac{2 z}{- 2 - i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = - \frac{1}{5} + \frac{i}{10} + \frac{2 z}{- 2 - i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = - \frac{1}{5} + \frac{i}{10} + \frac{2 z}{- 2 - i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.7

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

$y = - \frac{1}{5} + \frac{i}{10} + \frac{2 z}{- 2 - i} \cdot \frac{- 2 + i}{- 2 + i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8

Multiply.

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Step 11.3.3.3.1.8.1

Combine.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.2

Simplify the numerator.

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Step 11.3.3.3.1.8.2.1

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.2.2

Multiply $- 2$ by $2$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3

Simplify the denominator.

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Step 11.3.3.3.1.8.3.1

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

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Step 11.3.3.3.1.8.3.1.1

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.1.2

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.1.3

Apply the distributive property.

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{- 2 \cdot - 2 - 2 i - i \cdot - 2 - i i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{- 2 \cdot - 2 - 2 i - i \cdot - 2 - i i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.2

Simplify.

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Step 11.3.3.3.1.8.3.2.1

Multiply $- 2$ by $- 2$ .

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 - 2 i - i \cdot - 2 - i i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.2.2

Multiply $- 2$ by $- 1$ .

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 - 2 i + 2 i - i i} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.2.3

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.2.4

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.2.5

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

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 - 2 i + 2 i - i^{1 + 1}} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.2.6

Add $1$ and $1$ .

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 - 2 i + 2 i - i^{2}} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.2.7

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

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 + 0 - i^{2}} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.2.8

Add $4$ and $0$ .

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 - i^{2}} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 - i^{2}} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.3

Simplify each term.

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Step 11.3.3.3.1.8.3.3.1

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

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 + 1} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.3.2

Multiply $- 1$ by $- 1$ .

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 + 1} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{4 + 1} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.8.3.4

Add $4$ and $1$ .

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{5} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{5} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = - \frac{1}{5} + \frac{i}{10} + \frac{- 4 z + 2 z i}{5} + \frac{- 2 i z}{- 4 - 2 i}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.9

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

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Step 11.3.3.3.1.9.1

Factor $2 z$ out of $- 4 z$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.9.2

Factor $2 z$ out of $2 z i$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.9.3

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

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.10

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

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Step 11.3.3.3.1.10.1

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.10.2

Cancel the common factors.

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Step 11.3.3.3.1.10.2.1

Factor $2$ out of $- 4$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.10.2.2

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.10.2.3

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.10.2.4

Cancel the common factor.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.10.2.5

Rewrite the expression.

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.11

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12

Multiply.

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Step 11.3.3.3.1.12.1

Combine.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2

Simplify the numerator.

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Step 11.3.3.3.1.12.2.1

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.2

Multiply $- 2$ by $- 1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.3

Multiply $- i z i$ .

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Step 11.3.3.3.1.12.2.3.1

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.3.2

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.3.3

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.3.4

Add $1$ and $1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.4

Simplify each term.

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Step 11.3.3.3.1.12.2.4.1

Rewrite $- 1 z$ as $- z$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.4.2

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.4.3

Multiply $- z \cdot - 1$ .

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Step 11.3.3.3.1.12.2.4.3.1

Multiply $- 1$ by $- 1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.2.4.3.2

Multiply $z$ by $1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3

Simplify the denominator.

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Step 11.3.3.3.1.12.3.1

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

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Step 11.3.3.3.1.12.3.1.1

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.1.2

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.1.3

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.2

Simplify.

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Step 11.3.3.3.1.12.3.2.1

Multiply $- 2$ by $- 2$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.2.2

Multiply $- 2$ by $- 1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.2.3

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.2.4

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.2.5

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.2.6

Add $1$ and $1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.2.7

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

$y = - \frac{1}{5} + \frac{i}{10} + \frac{2 z (- 2 + i)}{5} + \frac{2 i z + z}{4 + 0 - i^{2}}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.2.8

Add $4$ and $0$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.3

Simplify each term.

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Step 11.3.3.3.1.12.3.3.1

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.3.2

Multiply $- 1$ by $- 1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.12.3.4

Add $4$ and $1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.13

Factor $z$ out of $2 i z + z$ .

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Step 11.3.3.3.1.13.1

Factor $z$ out of $2 i z$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.13.2

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.13.3

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

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.1.13.4

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

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.2

Combine the numerators over the common denominator.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.3

Simplify each term.

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Step 11.3.3.3.3.1

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.3.2

Multiply $- 2$ by $2$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.3.3

Apply the distributive property.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.3.4

Multiply $z$ by $1$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.3.5

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

$y = \frac{- 1 - 4 z + 2 z i + 2 z i + z}{5} + \frac{i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{- 1 - 4 z + 2 z i + 2 z i + z}{5} + \frac{i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.4

Simplify by adding terms.

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Step 11.3.3.3.4.1

Add $- 4 z$ and $z$ .

$y = \frac{- 1 + 2 z i + 2 z i - 3 z}{5} + \frac{i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.4.2

Add $2 z i$ and $2 z i$ .

$y = \frac{- 1 + 4 z i - 3 z}{5} + \frac{i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.4.3

Reorder terms.

$y = \frac{4 i z - 3 z - 1}{5} + \frac{i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{4 i z - 3 z - 1}{5} + \frac{i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.5

To write $\frac{4 i z - 3 z - 1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{4 i z - 3 z - 1}{5} \cdot \frac{2}{2} + \frac{i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.6

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.3.3.3.6.1

Multiply $\frac{4 i z - 3 z - 1}{5}$ by $\frac{2}{2}$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.6.2

Multiply $5$ by $2$ .

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

$y + i y - i z + z = 0$

$x = - y - z$

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.7

Combine the numerators over the common denominator.

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

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.8

Simplify the numerator.

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Step 11.3.3.3.8.1

Apply the distributive property.

$y = \frac{4 i z \cdot 2 - 3 z \cdot 2 - 1 \cdot 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.8.2

Simplify.

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Step 11.3.3.3.8.2.1

Multiply $2$ by $4$ .

$y = \frac{8 i z - 3 z \cdot 2 - 1 \cdot 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.8.2.2

Multiply $2$ by $- 3$ .

$y = \frac{8 i z - 6 z - 1 \cdot 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.3.3.3.8.2.3

Multiply $- 1$ by $2$ .

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$y + i y - i z + z = 0$

$x = - y - z$

Step 11.4

Replace all occurrences of $y$ with $\frac{8 i z - 6 z - 2 + i}{10}$ in each equation.

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Step 11.4.1

Replace all occurrences of $y$ in $y + i y - i z + z = 0$ with $\frac{8 i z - 6 z - 2 + i}{10}$ .

$(\frac{8 i z - 6 z - 2 + i}{10}) + i (\frac{8 i z - 6 z - 2 + i}{10}) - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2

Simplify the left side.

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Step 11.4.2.1

Simplify $(\frac{8 i z - 6 z - 2 + i}{10}) + i (\frac{8 i z - 6 z - 2 + i}{10}) - i z + z$ .

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Step 11.4.2.1.1

Simplify each term.

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Step 11.4.2.1.1.1

Combine $i$ and $\frac{8 i z - 6 z - 2 + i}{10}$ .

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{i (8 i z - 6 z - 2 + i)}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.2

Apply the distributive property.

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{i (8 i z) + i (- 6 z) + i \cdot - 2 + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3

Simplify.

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Step 11.4.2.1.1.3.1

Multiply $i (8 i z)$ .

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Step 11.4.2.1.1.3.1.1

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 (i i) z + i (- 6 z) + i \cdot - 2 + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3.1.2

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 (i i) z + i (- 6 z) + i \cdot - 2 + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3.1.3

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{1 + 1} z + i (- 6 z) + i \cdot - 2 + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3.1.4

Add $1$ and $1$ .

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) + i \cdot - 2 + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) + i \cdot - 2 + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3.2

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) - 2 \cdot i + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3.3

Multiply $i i$ .

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Step 11.4.2.1.1.3.3.1

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) - 2 \cdot i + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3.3.2

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) - 2 \cdot i + i i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3.3.3

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) - 2 \cdot i + i^{1 + 1}}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.3.3.4

Add $1$ and $1$ .

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) - 2 \cdot i + i^{2}}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) - 2 \cdot i + i^{2}}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 i^{2} z + i (- 6 z) - 2 \cdot i + i^{2}}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.4

Simplify each term.

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Step 11.4.2.1.1.4.1

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{8 \cdot (- 1 z) + i (- 6 z) - 2 \cdot i + i^{2}}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.4.2

Multiply $8$ by $- 1$ .

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{- 8 z + i (- 6 z) - 2 \cdot i + i^{2}}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.4.3

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

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{- 8 z + i (- 6 z) - 2 i - 1}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{- 8 z + i (- 6 z) - 2 i - 1}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.5

Reorder factors in $- 8 z + i (- 6 z) - 2 i - 1$ .

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{- 8 z - 6 i z - 2 i - 1}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.1.6

Reorder terms.

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{- 6 i z - 8 z - 1 - 2 i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$\frac{8 i z - 6 z - 2 + i}{10} + \frac{- 6 i z - 8 z - 1 - 2 i}{10} - i z + z = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.2

Simplify terms.

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Step 11.4.2.1.2.1

Combine the numerators over the common denominator.

$- i z + z + \frac{8 i z - 6 z - 2 + i - 6 i z - 8 z - 1 - 2 i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.2.2

Subtract $6 i z$ from $8 i z$ .

$- i z + z + \frac{2 i z - 6 z - 2 + i - 8 z - 1 - 2 i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.2.3

Subtract $8 z$ from $- 6 z$ .

$- i z + z + \frac{2 i z - 14 z - 2 + i - 1 - 2 i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.2.4

Subtract $1$ from $- 2$ .

$- i z + z + \frac{2 i z - 14 z - 3 + i - 2 i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.2.5

Subtract $2 i$ from $i$ .

$- i z + z + \frac{2 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$- i z + z + \frac{2 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.3

To write $- i z$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$z - i z \cdot \frac{10}{10} + \frac{2 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.4

Simplify terms.

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Step 11.4.2.1.4.1

Combine $- i z$ and $\frac{10}{10}$ .

$z + \frac{- i z \cdot 10}{10} + \frac{2 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.4.2

Combine the numerators over the common denominator.

$z + \frac{- i z \cdot 10 + 2 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$z + \frac{- i z \cdot 10 + 2 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.5

Simplify the numerator.

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Step 11.4.2.1.5.1

Multiply $10$ by $- 1$ .

$z + \frac{- 10 i z + 2 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.5.2

Add $- 10 i z$ and $2 i z$ .

$z + \frac{- 8 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$z + \frac{- 8 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.6

To write $z$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$z \cdot \frac{10}{10} + \frac{- 8 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.7

Simplify terms.

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Step 11.4.2.1.7.1

Combine $z$ and $\frac{10}{10}$ .

$\frac{z \cdot 10}{10} + \frac{- 8 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.7.2

Combine the numerators over the common denominator.

$\frac{z \cdot 10 - 8 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$\frac{z \cdot 10 - 8 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.8

Simplify the numerator.

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Step 11.4.2.1.8.1

Move $10$ to the left of $z$ .

$\frac{10 \cdot z - 8 i z - 14 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.8.2

Subtract $14 z$ from $10 z$ .

$\frac{- 8 i z - 4 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$\frac{- 8 i z - 4 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9

Simplify with factoring out.

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Step 11.4.2.1.9.1

Factor $- 1$ out of $- 8 i z$ .

$\frac{- (8 i z) - 4 z - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9.2

Factor $- 1$ out of $- 4 z$ .

$\frac{- (8 i z) - (4 z) - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9.3

Factor $- 1$ out of $- (8 i z) - (4 z)$ .

$\frac{- (8 i z + 4 z) - 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9.4

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

$\frac{- (8 i z + 4 z) - 1 \cdot 3 - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9.5

Factor $- 1$ out of $- (8 i z + 4 z) - 1 (3)$ .

$\frac{- (8 i z + 4 z + 3) - i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9.6

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

$\frac{- (8 i z + 4 z + 3) - (i)}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9.7

Factor $- 1$ out of $- (8 i z + 4 z + 3) - (i)$ .

$\frac{- (8 i z + 4 z + 3 + i)}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9.8

Simplify the expression.

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Step 11.4.2.1.9.8.1

Rewrite $- (8 i z + 4 z + 3 + i)$ as $- 1 (8 i z + 4 z + 3 + i)$ .

$\frac{- 1 (8 i z + 4 z + 3 + i)}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.2.1.9.8.2

Move the negative in front of the fraction.

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - y - z$

Step 11.4.3

Replace all occurrences of $y$ in $x = - y - z$ with $\frac{8 i z - 6 z - 2 + i}{10}$ .

$x = - (\frac{8 i z - 6 z - 2 + i}{10}) - z$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4

Simplify the right side.

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Step 11.4.4.1

Simplify $- (\frac{8 i z - 6 z - 2 + i}{10}) - z$ .

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Step 11.4.4.1.1

To write $- z$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$x = - \frac{8 i z - 6 z - 2 + i}{10} - z \cdot \frac{10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.2

Simplify terms.

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Step 11.4.4.1.2.1

Combine $- z$ and $\frac{10}{10}$ .

$x = - \frac{8 i z - 6 z - 2 + i}{10} + \frac{- z \cdot 10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.2.2

Combine the numerators over the common denominator.

$x = \frac{- (8 i z - 6 z - 2 + i) - z \cdot 10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = \frac{- (8 i z - 6 z - 2 + i) - z \cdot 10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.3

Simplify the numerator.

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Step 11.4.4.1.3.1

Apply the distributive property.

$x = \frac{- (8 i z) - (- 6 z) + 2 - i - z \cdot 10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.3.2

Simplify.

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Step 11.4.4.1.3.2.1

Multiply $8$ by $- 1$ .

$x = \frac{- 8 (i z) - (- 6 z) + 2 - i - z \cdot 10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.3.2.2

Multiply $- 6$ by $- 1$ .

$x = \frac{- 8 (i z) + 6 z + 2 - i - z \cdot 10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.3.2.3

Multiply $- 1$ by $- 2$ .

$x = \frac{- 8 (i z) + 6 z + 2 - i - z \cdot 10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = \frac{- 8 (i z) + 6 z + 2 - i - z \cdot 10}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.3.3

Multiply $10$ by $- 1$ .

$x = \frac{- 8 i z + 6 z + 2 - i - 10 z}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.3.4

Subtract $10 z$ from $6 z$ .

$x = \frac{- 8 i z - 4 z + 2 - i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = \frac{- 8 i z - 4 z + 2 - i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4

Simplify with factoring out.

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Step 11.4.4.1.4.1

Factor $- 1$ out of $- 8 i z$ .

$x = \frac{- (8 i z) - 4 z + 2 - i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4.2

Factor $- 1$ out of $- 4 z$ .

$x = \frac{- (8 i z) - (4 z) + 2 - i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4.3

Factor $- 1$ out of $- (8 i z) - (4 z)$ .

$x = \frac{- (8 i z + 4 z) + 2 - i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4.4

Rewrite $2$ as $- 1 (- 2)$ .

$x = \frac{- (8 i z + 4 z) - 1 \cdot - 2 - i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4.5

Factor $- 1$ out of $- (8 i z + 4 z) - 1 (- 2)$ .

$x = \frac{- (8 i z + 4 z - 2) - i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4.6

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

$x = \frac{- (8 i z + 4 z - 2) - (i)}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4.7

Factor $- 1$ out of $- (8 i z + 4 z - 2) - (i)$ .

$x = \frac{- (8 i z + 4 z - 2 + i)}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4.8

Simplify the expression.

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Step 11.4.4.1.4.8.1

Rewrite $- (8 i z + 4 z - 2 + i)$ as $- 1 (8 i z + 4 z - 2 + i)$ .

$x = \frac{- 1 (8 i z + 4 z - 2 + i)}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.4.4.1.4.8.2

Move the negative in front of the fraction.

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$- \frac{8 i z + 4 z + 3 + i}{10} = 0$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5

Solve for $z$ in $- \frac{8 i z + 4 z + 3 + i}{10} = 0$ .

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Step 11.5.1

Set the numerator equal to zero.

$8 i z + 4 z + 3 + i = 0$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2

Solve the equation for $z$ .

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Step 11.5.2.1

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

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Step 11.5.2.1.1

Subtract $3$ from both sides of the equation.

$8 i z + 4 z + i = - 3$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.1.2

Subtract $i$ from both sides of the equation.

$8 i z + 4 z = - 3 - i$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$8 i z + 4 z = - 3 - i$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.2

Factor $4 z$ out of $8 i z + 4 z$ .

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Step 11.5.2.2.1

Factor $4 z$ out of $8 i z$ .

$4 z (2 i) + 4 z = - 3 - i$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.2.2

Factor $4 z$ out of $4 z$ .

$4 z (2 i) + 4 z (1) = - 3 - i$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.2.3

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

$4 z (2 i + 1) = - 3 - i$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$4 z (2 i + 1) = - 3 - i$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3

Divide each term in $4 z (2 i + 1) = - 3 - i$ by $8 i + 4$ and simplify.

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Step 11.5.2.3.1

Divide each term in $4 z (2 i + 1) = - 3 - i$ by $8 i + 4$ .

$\frac{4 z (2 i + 1)}{8 i + 4} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.2

Simplify the left side.

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Step 11.5.2.3.2.1

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

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Step 11.5.2.3.2.1.1

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

$\frac{4 (z (2 i + 1))}{8 i + 4} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.2.1.2

Cancel the common factors.

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Step 11.5.2.3.2.1.2.1

Factor $4$ out of $8 i$ .

$\frac{4 (z (2 i + 1))}{4 (2 i) + 4} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.2.1.2.2

Factor $4$ out of $4$ .

$\frac{4 (z (2 i + 1))}{4 (2 i) + 4 (1)} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.2.1.2.3

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

$\frac{4 (z (2 i + 1))}{4 (2 i + 1)} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.2.1.2.4

Cancel the common factor.

$\frac{4 (z (2 i + 1))}{4 (2 i + 1)} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.2.1.2.5

Rewrite the expression.

$\frac{z (2 i + 1)}{2 i + 1} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$\frac{z (2 i + 1)}{2 i + 1} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$\frac{z (2 i + 1)}{2 i + 1} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.2.2

Cancel the common factor of $2 i + 1$ .

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Step 11.5.2.3.2.2.1

Cancel the common factor.

$\frac{z (2 i + 1)}{2 i + 1} = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.2.2.2

Divide $z$ by $1$ .

$z = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 3}{8 i + 4} + \frac{- i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3

Simplify the right side.

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Step 11.5.2.3.3.1

Combine the numerators over the common denominator.

$z = \frac{- 3 - i}{8 i + 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.2

Multiply the numerator and denominator of $\frac{- 3 - i}{4 + 8 i}$ by the conjugate of $4 + 8 i$ to make the denominator real.

$z = \frac{- 3 - i}{4 + 8 i} \cdot \frac{4 - 8 i}{4 - 8 i}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3

Multiply.

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Step 11.5.2.3.3.3.1

Combine.

$z = \frac{(- 3 - i) (4 - 8 i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2

Simplify the numerator.

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Step 11.5.2.3.3.3.2.1

Expand $(- 3 - i) (4 - 8 i)$ using the FOIL Method.

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Step 11.5.2.3.3.3.2.1.1

Apply the distributive property.

$z = \frac{- 3 (4 - 8 i) - i (4 - 8 i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.1.2

Apply the distributive property.

$z = \frac{- 3 \cdot 4 - 3 (- 8 i) - i (4 - 8 i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.1.3

Apply the distributive property.

$z = \frac{- 3 \cdot 4 - 3 (- 8 i) - i \cdot 4 - i (- 8 i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 3 \cdot 4 - 3 (- 8 i) - i \cdot 4 - i (- 8 i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2

Simplify and combine like terms.

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Step 11.5.2.3.3.3.2.2.1

Simplify each term.

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Step 11.5.2.3.3.3.2.2.1.1

Multiply $- 3$ by $4$ .

$z = \frac{- 12 - 3 (- 8 i) - i \cdot 4 - i (- 8 i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.2

Multiply $- 8$ by $- 3$ .

$z = \frac{- 12 + 24 i - i \cdot 4 - i (- 8 i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.3

Multiply $4$ by $- 1$ .

$z = \frac{- 12 + 24 i - 4 i - i (- 8 i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.4

Multiply $- i (- 8 i)$ .

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Step 11.5.2.3.3.3.2.2.1.4.1

Multiply $- 8$ by $- 1$ .

$z = \frac{- 12 + 24 i - 4 i + 8 i i}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.4.2

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

$z = \frac{- 12 + 24 i - 4 i + 8 (i i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.4.3

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

$z = \frac{- 12 + 24 i - 4 i + 8 (i i)}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.4.4

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

$z = \frac{- 12 + 24 i - 4 i + 8 i^{1 + 1}}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.4.5

Add $1$ and $1$ .

$z = \frac{- 12 + 24 i - 4 i + 8 i^{2}}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 12 + 24 i - 4 i + 8 i^{2}}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.5

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

$z = \frac{- 12 + 24 i - 4 i + 8 \cdot - 1}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.1.6

Multiply $8$ by $- 1$ .

$z = \frac{- 12 + 24 i - 4 i - 8}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 12 + 24 i - 4 i - 8}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.2

Subtract $8$ from $- 12$ .

$z = \frac{- 20 + 24 i - 4 i}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.2.2.3

Subtract $4 i$ from $24 i$ .

$z = \frac{- 20 + 20 i}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 20 + 20 i}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 20 + 20 i}{(4 + 8 i) (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3

Simplify the denominator.

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Step 11.5.2.3.3.3.3.1

Expand $(4 + 8 i) (4 - 8 i)$ using the FOIL Method.

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Step 11.5.2.3.3.3.3.1.1

Apply the distributive property.

$z = \frac{- 20 + 20 i}{4 (4 - 8 i) + 8 i (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.1.2

Apply the distributive property.

$z = \frac{- 20 + 20 i}{4 \cdot 4 + 4 (- 8 i) + 8 i (4 - 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.1.3

Apply the distributive property.

$z = \frac{- 20 + 20 i}{4 \cdot 4 + 4 (- 8 i) + 8 i \cdot 4 + 8 i (- 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 20 + 20 i}{4 \cdot 4 + 4 (- 8 i) + 8 i \cdot 4 + 8 i (- 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2

Simplify.

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Step 11.5.2.3.3.3.3.2.1

Multiply $4$ by $4$ .

$z = \frac{- 20 + 20 i}{16 + 4 (- 8 i) + 8 i \cdot 4 + 8 i (- 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.2

Multiply $- 8$ by $4$ .

$z = \frac{- 20 + 20 i}{16 - 32 i + 8 i \cdot 4 + 8 i (- 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.3

Multiply $4$ by $8$ .

$z = \frac{- 20 + 20 i}{16 - 32 i + 32 i + 8 i (- 8 i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.4

Multiply $- 8$ by $8$ .

$z = \frac{- 20 + 20 i}{16 - 32 i + 32 i - 64 i i}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.5

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

$z = \frac{- 20 + 20 i}{16 - 32 i + 32 i - 64 (i i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.6

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

$z = \frac{- 20 + 20 i}{16 - 32 i + 32 i - 64 (i i)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.7

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

$z = \frac{- 20 + 20 i}{16 - 32 i + 32 i - 64 i^{1 + 1}}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.8

Add $1$ and $1$ .

$z = \frac{- 20 + 20 i}{16 - 32 i + 32 i - 64 i^{2}}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.9

Add $- 32 i$ and $32 i$ .

$z = \frac{- 20 + 20 i}{16 + 0 - 64 i^{2}}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.2.10

Add $16$ and $0$ .

$z = \frac{- 20 + 20 i}{16 - 64 i^{2}}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 20 + 20 i}{16 - 64 i^{2}}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.3

Simplify each term.

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Step 11.5.2.3.3.3.3.3.1

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

$z = \frac{- 20 + 20 i}{16 - 64 \cdot - 1}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.3.2

Multiply $- 64$ by $- 1$ .

$z = \frac{- 20 + 20 i}{16 + 64}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 20 + 20 i}{16 + 64}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.3.3.4

Add $16$ and $64$ .

$z = \frac{- 20 + 20 i}{80}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 20 + 20 i}{80}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 20 + 20 i}{80}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.4

Cancel the common factor of $- 20 + 20 i$ and $80$ .

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Step 11.5.2.3.3.4.1

Factor $20$ out of $- 20$ .

$z = \frac{20 \cdot - 1 + 20 i}{80}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.4.2

Factor $20$ out of $20 i$ .

$z = \frac{20 \cdot - 1 + 20 (i)}{80}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.4.3

Factor $20$ out of $20 \cdot - 1 + 20 (i)$ .

$z = \frac{20 \cdot (- 1 + i)}{80}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.4.4

Cancel the common factors.

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Step 11.5.2.3.3.4.4.1

Factor $20$ out of $80$ .

$z = \frac{20 \cdot (- 1 + i)}{20 (4)}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.4.4.2

Cancel the common factor.

$z = \frac{20 \cdot (- 1 + i)}{20 \cdot 4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.4.4.3

Rewrite the expression.

$z = \frac{- 1 + i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 1 + i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = \frac{- 1 + i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.5

Split the fraction $\frac{- 1 + i}{4}$ into two fractions.

$z = \frac{- 1}{4} + \frac{i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.5.2.3.3.6

Move the negative in front of the fraction.

$z = - \frac{1}{4} + \frac{i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = - \frac{1}{4} + \frac{i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = - \frac{1}{4} + \frac{i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = - \frac{1}{4} + \frac{i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

$z = - \frac{1}{4} + \frac{i}{4}$

$x = - \frac{8 i z + 4 z - 2 + i}{10}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.6

Replace all occurrences of $z$ with $- \frac{1}{4} + \frac{i}{4}$ in each equation.

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Step 11.6.1

Replace all occurrences of $z$ in $x = - \frac{8 i z + 4 z - 2 + i}{10}$ with $- \frac{1}{4} + \frac{i}{4}$ .

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

$z = - \frac{1}{4} + \frac{i}{4}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.6.2

Simplify the right side.

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Step 11.6.2.1

Reorder $8 i (- \frac{1}{4} + \frac{i}{4}) + 4 (- \frac{1}{4} + \frac{i}{4})$ and $- 2$ .

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

$z = - \frac{1}{4} + \frac{i}{4}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

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

$z = - \frac{1}{4} + \frac{i}{4}$

$y = \frac{8 i z - 6 z - 2 + i}{10}$

Step 11.6.3

Replace all occurrences of $z$ in $y = \frac{8 i z - 6 z - 2 + i}{10}$ with $- \frac{1}{4} + \frac{i}{4}$ .

$y = \frac{8 i (- \frac{1}{4} + \frac{i}{4}) - 6 (- \frac{1}{4} + \frac{i}{4}) - 2 + i}{10}$

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

$z = - \frac{1}{4} + \frac{i}{4}$

Step 11.6.4

Simplify the right side.

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Step 11.6.4.1

Reorder $8 i (- \frac{1}{4} + \frac{i}{4}) - 6 (- \frac{1}{4} + \frac{i}{4})$ and $- 2$ .

$y = \frac{- 2 + 8 i (- \frac{1}{4} + \frac{i}{4}) - 6 (- \frac{1}{4} + \frac{i}{4}) + i}{10}$

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

$z = - \frac{1}{4} + \frac{i}{4}$

$y = \frac{- 2 + 8 i (- \frac{1}{4} + \frac{i}{4}) - 6 (- \frac{1}{4} + \frac{i}{4}) + i}{10}$

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

$z = - \frac{1}{4} + \frac{i}{4}$

$y = \frac{- 2 + 8 i (- \frac{1}{4} + \frac{i}{4}) - 6 (- \frac{1}{4} + \frac{i}{4}) + i}{10}$

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

$z = - \frac{1}{4} + \frac{i}{4}$

Step 11.7

List all of the solutions.

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