Finite Math Examples

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

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

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

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Subtract $\frac{{(1 + i)}^{2}}{{(1 - i)}^{2}}$ from both sides of the equation.

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

Step 2.2.2

Simplify each term.

Tap for more steps...

Step 2.2.2.1

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

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

Step 2.2.2.2

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

Tap for more steps...

Step 2.2.2.2.1

Apply the distributive property.

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

Step 2.2.2.2.2

Apply the distributive property.

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

Step 2.2.2.2.3

Apply the distributive property.

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

Step 2.2.2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.2.2.3.1.1

Multiply $1$ by $1$ .

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

Step 2.2.2.3.1.2

Multiply $i$ by $1$ .

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

Step 2.2.2.3.1.3

Multiply $i$ by $1$ .

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

Step 2.2.2.3.1.4

Multiply $i i$ .

Tap for more steps...

Step 2.2.2.3.1.4.1

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

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

Step 2.2.2.3.1.4.2

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

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

Step 2.2.2.3.1.4.3

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

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

Step 2.2.2.3.1.4.4

Add $1$ and $1$ .

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

Step 2.2.2.3.1.5

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

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

Step 2.2.2.3.2

Subtract $1$ from $1$ .

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

Step 2.2.2.3.3

Add $0$ and $i$ .

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

Step 2.2.2.3.4

Add $i$ and $i$ .

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

Step 2.2.2.4

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

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

Step 2.2.2.5

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

Tap for more steps...

Step 2.2.2.5.1

Apply the distributive property.

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

Step 2.2.2.5.2

Apply the distributive property.

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

Step 2.2.2.5.3

Apply the distributive property.

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

Step 2.2.2.6

Simplify and combine like terms.

Tap for more steps...

Step 2.2.2.6.1

Simplify each term.

Tap for more steps...

Step 2.2.2.6.1.1

Multiply $1$ by $1$ .

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

Step 2.2.2.6.1.2

Multiply $- i$ by $1$ .

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

Step 2.2.2.6.1.3

Multiply $- 1$ by $1$ .

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

Step 2.2.2.6.1.4

Multiply $- i (- i)$ .

Tap for more steps...

Step 2.2.2.6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2.6.1.4.2

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

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

Step 2.2.2.6.1.4.3

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

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

Step 2.2.2.6.1.4.4

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

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

Step 2.2.2.6.1.4.5

Add $1$ and $1$ .

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

Step 2.2.2.6.1.4.6

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

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

Step 2.2.2.6.1.5

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

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

Step 2.2.2.6.2

Subtract $1$ from $1$ .

$\frac{2}{x + i y} = 2 + 2 i - \frac{2 i}{0 - i - i}$

Step 2.2.2.6.3

Subtract $i$ from $0$ .

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

Step 2.2.2.6.4

Subtract $i$ from $- i$ .

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

Step 2.2.2.7

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

Tap for more steps...

Step 2.2.2.7.1

Factor $2$ out of $2 i$ .

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

Step 2.2.2.7.2

Cancel the common factors.

Tap for more steps...

Step 2.2.2.7.2.1

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

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

Step 2.2.2.7.2.2

Cancel the common factor.

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

Step 2.2.2.7.2.3

Rewrite the expression.

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

Step 2.2.2.8

Cancel the common factor of $i$ .

Tap for more steps...

Step 2.2.2.8.1

Cancel the common factor.

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

Step 2.2.2.8.2

Rewrite the expression.

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

Step 2.2.2.8.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

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

Step 2.2.2.9

Multiply $- (- 1 \cdot 1)$ .

Tap for more steps...

Step 2.2.2.9.1

Multiply $- 1$ by $1$ .

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

Step 2.2.2.9.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.3

Add $2$ and $1$ .

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

Step 2.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x + i y, 1, 1$

Step 2.3.2

Remove parentheses.

$x + i y, 1, 1$

Step 2.3.3

The LCM of one and any expression is the expression.

$x + i y$

Step 2.4

Multiply each term in $\frac{2}{x + i y} = 3 + 2 i$ by $x + i y$ to eliminate the fractions.

Tap for more steps...

Step 2.4.1

Multiply each term in $\frac{2}{x + i y} = 3 + 2 i$ by $x + i y$ .

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

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $x + i y$ .

Tap for more steps...

Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Rewrite the expression.

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

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Simplify each term.

Tap for more steps...

Step 2.4.3.1.1

Apply the distributive property.

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

Step 2.4.3.1.2

Apply the distributive property.

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

Step 2.4.3.1.3

Multiply $2 i (i y)$ .

Tap for more steps...

Step 2.4.3.1.3.1

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

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

Step 2.4.3.1.3.2

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

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

Step 2.4.3.1.3.3

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

$2 = 3 x + 3 i y + 2 i x + 2 i^{1 + 1} y$

Step 2.4.3.1.3.4

Add $1$ and $1$ .

$2 = 3 x + 3 i y + 2 i x + 2 i^{2} y$

Step 2.4.3.1.4

Simplify each term.

Tap for more steps...

Step 2.4.3.1.4.1

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

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

Step 2.4.3.1.4.2

Multiply $2$ by $- 1$ .

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

Step 2.5

Solve the equation.

Tap for more steps...

Step 2.5.1

Rewrite the equation as $3 x + 3 i y + 2 i x - 2 y = 2$ .

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

Step 2.5.2

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

Tap for more steps...

Step 2.5.2.1

Subtract $3 i y$ from both sides of the equation.

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

Step 2.5.2.2

Add $2 y$ to both sides of the equation.

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

Step 2.5.3

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

Tap for more steps...

Step 2.5.3.1

Factor $x$ out of $3 x$ .

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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

Step 2.5.4

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

Tap for more steps...

Step 2.5.4.1

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

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

Step 2.5.4.2

Simplify the left side.

Tap for more steps...

Step 2.5.4.2.1

Cancel the common factor of $3 + 2 i$ .

Tap for more steps...

Step 2.5.4.2.1.1

Cancel the common factor.

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

Step 2.5.4.2.1.2

Divide $x$ by $1$ .

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

Step 2.5.4.3

Simplify the right side.

Tap for more steps...

Step 2.5.4.3.1

Simplify each term.

Tap for more steps...

Step 2.5.4.3.1.1

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

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

Step 2.5.4.3.1.2

Multiply.

Tap for more steps...

Step 2.5.4.3.1.2.1

Combine.

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

Step 2.5.4.3.1.2.2

Simplify the numerator.

Tap for more steps...

Step 2.5.4.3.1.2.2.1

Apply the distributive property.

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

Step 2.5.4.3.1.2.2.2

Multiply $2$ by $3$ .

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

Step 2.5.4.3.1.2.2.3

Multiply $- 2$ by $2$ .

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

Step 2.5.4.3.1.2.3

Simplify the denominator.

Tap for more steps...

Step 2.5.4.3.1.2.3.1

Expand $(3 + 2 i) (3 - 2 i)$ using the FOIL Method.

Tap for more steps...

Step 2.5.4.3.1.2.3.1.1

Apply the distributive property.

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

Step 2.5.4.3.1.2.3.1.2

Apply the distributive property.

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

Step 2.5.4.3.1.2.3.1.3

Apply the distributive property.

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

Step 2.5.4.3.1.2.3.2

Simplify.

Tap for more steps...

Step 2.5.4.3.1.2.3.2.1

Multiply $3$ by $3$ .

$x = \frac{6 - 4 i}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.2

Multiply $- 2$ by $3$ .

$x = \frac{6 - 4 i}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.3

Multiply $3$ by $2$ .

$x = \frac{6 - 4 i}{9 - 6 i + 6 i + 2 i (- 2 i)} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.4

Multiply $- 2$ by $2$ .

$x = \frac{6 - 4 i}{9 - 6 i + 6 i - 4 i i} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.5

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

$x = \frac{6 - 4 i}{9 - 6 i + 6 i - 4 (i^{1} i)} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.6

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

$x = \frac{6 - 4 i}{9 - 6 i + 6 i - 4 (i^{1} i^{1})} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.7

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

$x = \frac{6 - 4 i}{9 - 6 i + 6 i - 4 i^{1 + 1}} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.8

Add $1$ and $1$ .

$x = \frac{6 - 4 i}{9 - 6 i + 6 i - 4 i^{2}} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.9

Add $- 6 i$ and $6 i$ .

$x = \frac{6 - 4 i}{9 + 0 - 4 i^{2}} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.2.10

Add $9$ and $0$ .

$x = \frac{6 - 4 i}{9 - 4 i^{2}} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.3

Simplify each term.

Tap for more steps...

Step 2.5.4.3.1.2.3.3.1

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

$x = \frac{6 - 4 i}{9 - 4 \cdot - 1} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.3.2

Multiply $- 4$ by $- 1$ .

$x = \frac{6 - 4 i}{9 + 4} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.2.3.4

Add $9$ and $4$ .

$x = \frac{6 - 4 i}{13} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.3

Split the fraction $\frac{6 - 4 i}{13}$ into two fractions.

$x = \frac{6}{13} + \frac{- 4 i}{13} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.4

Move the negative in front of the fraction.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 3 i y}{3 + 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.5

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 3 i y}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6

Multiply.

Tap for more steps...

Step 2.5.4.3.1.6.1

Combine.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 3 i y (3 - 2 i)}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2

Simplify the numerator.

Tap for more steps...

Step 2.5.4.3.1.6.2.1

Apply the distributive property.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 3 i y \cdot 3 - 3 i y (- 2 i)}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2.2

Multiply $3$ by $- 3$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 3 i y (- 2 i)}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2.3

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

Tap for more steps...

Step 2.5.4.3.1.6.2.3.1

Multiply $- 2$ by $- 3$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y + 6 i y i}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2.3.2

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y + 6 y (i^{1} i)}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2.3.3

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y + 6 y (i^{1} i^{1})}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2.3.4

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y + 6 y i^{1 + 1}}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2.3.5

Add $1$ and $1$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y + 6 y i^{2}}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2.4

Simplify each term.

Tap for more steps...

Step 2.5.4.3.1.6.2.4.1

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y + 6 y \cdot - 1}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.2.4.2

Multiply $- 1$ by $6$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{(3 + 2 i) (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3

Simplify the denominator.

Tap for more steps...

Step 2.5.4.3.1.6.3.1

Expand $(3 + 2 i) (3 - 2 i)$ using the FOIL Method.

Tap for more steps...

Step 2.5.4.3.1.6.3.1.1

Apply the distributive property.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{3 (3 - 2 i) + 2 i (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.1.2

Apply the distributive property.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.1.3

Apply the distributive property.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2

Simplify.

Tap for more steps...

Step 2.5.4.3.1.6.3.2.1

Multiply $3$ by $3$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.2

Multiply $- 2$ by $3$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.3

Multiply $3$ by $2$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 6 i + 6 i + 2 i (- 2 i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.4

Multiply $- 2$ by $2$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 6 i + 6 i - 4 i i} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.5

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 6 i + 6 i - 4 (i^{1} i)} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.6

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 6 i + 6 i - 4 (i^{1} i^{1})} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.7

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 6 i + 6 i - 4 i^{1 + 1}} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.8

Add $1$ and $1$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 6 i + 6 i - 4 i^{2}} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.9

Add $- 6 i$ and $6 i$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 + 0 - 4 i^{2}} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.2.10

Add $9$ and $0$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 4 i^{2}} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.3

Simplify each term.

Tap for more steps...

Step 2.5.4.3.1.6.3.3.1

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 - 4 \cdot - 1} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.3.2

Multiply $- 4$ by $- 1$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{9 + 4} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.6.3.4

Add $9$ and $4$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{- 9 i y - 6 y}{13} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.7

Factor $3 y$ out of $- 9 i y - 6 y$ .

Tap for more steps...

Step 2.5.4.3.1.7.1

Factor $3 y$ out of $- 9 i y$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i) - 6 y}{13} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.7.2

Factor $3 y$ out of $- 6 y$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i) + 3 y (- 2)}{13} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.7.3

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{2 y}{3 + 2 i}$

Step 2.5.4.3.1.8

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{2 y}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}$

Step 2.5.4.3.1.9

Multiply.

Tap for more steps...

Step 2.5.4.3.1.9.1

Combine.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{2 y (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.5.4.3.1.9.2

Simplify the numerator.

Tap for more steps...

Step 2.5.4.3.1.9.2.1

Apply the distributive property.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{2 y \cdot 3 + 2 y (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.5.4.3.1.9.2.2

Multiply $3$ by $2$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y + 2 y (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.5.4.3.1.9.2.3

Multiply $- 2$ by $2$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{(3 + 2 i) (3 - 2 i)}$

Step 2.5.4.3.1.9.3

Simplify the denominator.

Tap for more steps...

Step 2.5.4.3.1.9.3.1

Expand $(3 + 2 i) (3 - 2 i)$ using the FOIL Method.

Tap for more steps...

Step 2.5.4.3.1.9.3.1.1

Apply the distributive property.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{3 (3 - 2 i) + 2 i (3 - 2 i)}$

Step 2.5.4.3.1.9.3.1.2

Apply the distributive property.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)}$

Step 2.5.4.3.1.9.3.1.3

Apply the distributive property.

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.5.4.3.1.9.3.2

Simplify.

Tap for more steps...

Step 2.5.4.3.1.9.3.2.1

Multiply $3$ by $3$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.5.4.3.1.9.3.2.2

Multiply $- 2$ by $3$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.5.4.3.1.9.3.2.3

Multiply $3$ by $2$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 6 i + 6 i + 2 i (- 2 i)}$

Step 2.5.4.3.1.9.3.2.4

Multiply $- 2$ by $2$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 6 i + 6 i - 4 i i}$

Step 2.5.4.3.1.9.3.2.5

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 6 i + 6 i - 4 (i^{1} i)}$

Step 2.5.4.3.1.9.3.2.6

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 6 i + 6 i - 4 (i^{1} i^{1})}$

Step 2.5.4.3.1.9.3.2.7

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 6 i + 6 i - 4 i^{1 + 1}}$

Step 2.5.4.3.1.9.3.2.8

Add $1$ and $1$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 6 i + 6 i - 4 i^{2}}$

Step 2.5.4.3.1.9.3.2.9

Add $- 6 i$ and $6 i$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 + 0 - 4 i^{2}}$

Step 2.5.4.3.1.9.3.2.10

Add $9$ and $0$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 4 i^{2}}$

Step 2.5.4.3.1.9.3.3

Simplify each term.

Tap for more steps...

Step 2.5.4.3.1.9.3.3.1

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 - 4 \cdot - 1}$

Step 2.5.4.3.1.9.3.3.2

Multiply $- 4$ by $- 1$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{9 + 4}$

Step 2.5.4.3.1.9.3.4

Add $9$ and $4$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{6 y - 4 y i}{13}$

Step 2.5.4.3.1.10

Factor $2 y$ out of $6 y - 4 y i$ .

Tap for more steps...

Step 2.5.4.3.1.10.1

Factor $2 y$ out of $6 y$ .

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{2 y (3) - 4 y i}{13}$

Step 2.5.4.3.1.10.2

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{2 y (3) + 2 y (- 2 i)}{13}$

Step 2.5.4.3.1.10.3

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

$x = \frac{6}{13} - \frac{4 i}{13} + \frac{3 y (- 3 i - 2)}{13} + \frac{2 y (3 - 2 i)}{13}$

Step 2.5.4.3.2

Combine the numerators over the common denominator.

$x = \frac{6 - 4 i + 3 y (- 3 i - 2) + 2 y (3 - 2 i)}{13}$

Step 2.5.4.3.3

Simplify each term.

Tap for more steps...

Step 2.5.4.3.3.1

Apply the distributive property.

$x = \frac{6 - 4 i + 3 y (- 3 i) + 3 y \cdot - 2 + 2 y (3 - 2 i)}{13}$

Step 2.5.4.3.3.2

Multiply $- 3$ by $3$ .

$x = \frac{6 - 4 i - 9 y i + 3 y \cdot - 2 + 2 y (3 - 2 i)}{13}$

Step 2.5.4.3.3.3

Multiply $- 2$ by $3$ .

$x = \frac{6 - 4 i - 9 y i - 6 y + 2 y (3 - 2 i)}{13}$

Step 2.5.4.3.3.4

Apply the distributive property.

$x = \frac{6 - 4 i - 9 y i - 6 y + 2 y \cdot 3 + 2 y (- 2 i)}{13}$

Step 2.5.4.3.3.5

Multiply $3$ by $2$ .

$x = \frac{6 - 4 i - 9 y i - 6 y + 6 y + 2 y (- 2 i)}{13}$

Step 2.5.4.3.3.6

Multiply $- 2$ by $2$ .

$x = \frac{6 - 4 i - 9 y i - 6 y + 6 y - 4 y i}{13}$

Step 2.5.4.3.4

Simplify terms.

Tap for more steps...

Step 2.5.4.3.4.1

Combine the opposite terms in $6 - 4 i - 9 y i - 6 y + 6 y - 4 y i$ .

Tap for more steps...

Step 2.5.4.3.4.1.1

Add $- 6 y$ and $6 y$ .

$x = \frac{6 - 4 i - 9 y i + 0 - 4 y i}{13}$

Step 2.5.4.3.4.1.2

Add $6 - 4 i - 9 y i$ and $0$ .

$x = \frac{6 - 4 i - 9 y i - 4 y i}{13}$

Step 2.5.4.3.4.2

Subtract $4 y i$ from $- 9 y i$ .

$x = \frac{6 - 4 i - 13 y i}{13}$

Step 2.5.4.3.4.3

Reorder terms.

$x = \frac{- 13 i y + 6 - 4 i}{13}$

Step 2.5.4.3.4.4

Factor $- 1$ out of $- 13 i y$ .

$x = \frac{- (13 i y) + 6 - 4 i}{13}$

Step 2.5.4.3.4.5

Rewrite $6$ as $- 1 (- 6)$ .

$x = \frac{- (13 i y) - 1 (- 6) - 4 i}{13}$

Step 2.5.4.3.4.6

Factor $- 1$ out of $- (13 i y) - 1 (- 6)$ .

$x = \frac{- (13 i y - 6) - 4 i}{13}$

Step 2.5.4.3.4.7

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

$x = \frac{- (13 i y - 6) - (4 i)}{13}$

Step 2.5.4.3.4.8

Factor $- 1$ out of $- (13 i y - 6) - (4 i)$ .

$x = \frac{- (13 i y - 6 + 4 i)}{13}$

Step 2.5.4.3.4.9

Simplify the expression.

Tap for more steps...

Step 2.5.4.3.4.9.1

Rewrite $- (13 i y - 6 + 4 i)$ as $- 1 (13 i y - 6 + 4 i)$ .

$x = \frac{- 1 (13 i y - 6 + 4 i)}{13}$

Step 2.5.4.3.4.9.2

Move the negative in front of the fraction.

$x = - \frac{13 i y - 6 + 4 i}{13}$

Step 2.6

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 2.7

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

Tap for more steps...

Step 2.7.1

Subtract $\frac{{(1 + i)}^{2}}{{(1 - i)}^{2}}$ from both sides of the equation.

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

Step 2.7.2

Simplify each term.

Tap for more steps...

Step 2.7.2.1

Apply the distributive property.

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

Step 2.7.2.2

Multiply $- 1$ by $2$ .

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

Step 2.7.2.3

Multiply $2$ by $- 1$ .

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

Step 2.7.2.4

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

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

Step 2.7.2.5

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

Tap for more steps...

Step 2.7.2.5.1

Apply the distributive property.

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

Step 2.7.2.5.2

Apply the distributive property.

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

Step 2.7.2.5.3

Apply the distributive property.

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

Step 2.7.2.6

Simplify and combine like terms.

Tap for more steps...

Step 2.7.2.6.1

Simplify each term.

Tap for more steps...

Step 2.7.2.6.1.1

Multiply $1$ by $1$ .

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

Step 2.7.2.6.1.2

Multiply $i$ by $1$ .

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

Step 2.7.2.6.1.3

Multiply $i$ by $1$ .

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

Step 2.7.2.6.1.4

Multiply $i i$ .

Tap for more steps...

Step 2.7.2.6.1.4.1

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

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

Step 2.7.2.6.1.4.2

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

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

Step 2.7.2.6.1.4.3

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

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

Step 2.7.2.6.1.4.4

Add $1$ and $1$ .

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

Step 2.7.2.6.1.5

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

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

Step 2.7.2.6.2

Subtract $1$ from $1$ .

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

Step 2.7.2.6.3

Add $0$ and $i$ .

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

Step 2.7.2.6.4

Add $i$ and $i$ .

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

Step 2.7.2.7

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

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

Step 2.7.2.8

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

Tap for more steps...

Step 2.7.2.8.1

Apply the distributive property.

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

Step 2.7.2.8.2

Apply the distributive property.

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

Step 2.7.2.8.3

Apply the distributive property.

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

Step 2.7.2.9

Simplify and combine like terms.

Tap for more steps...

Step 2.7.2.9.1

Simplify each term.

Tap for more steps...

Step 2.7.2.9.1.1

Multiply $1$ by $1$ .

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

Step 2.7.2.9.1.2

Multiply $- i$ by $1$ .

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

Step 2.7.2.9.1.3

Multiply $- 1$ by $1$ .

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

Step 2.7.2.9.1.4

Multiply $- i (- i)$ .

Tap for more steps...

Step 2.7.2.9.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.7.2.9.1.4.2

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

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

Step 2.7.2.9.1.4.3

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

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

Step 2.7.2.9.1.4.4

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

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

Step 2.7.2.9.1.4.5

Add $1$ and $1$ .

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

Step 2.7.2.9.1.4.6

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

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

Step 2.7.2.9.1.5

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

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

Step 2.7.2.9.2

Subtract $1$ from $1$ .

$\frac{2}{x + i y} = - 2 - 2 i - \frac{2 i}{0 - i - i}$

Step 2.7.2.9.3

Subtract $i$ from $0$ .

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

Step 2.7.2.9.4

Subtract $i$ from $- i$ .

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

Step 2.7.2.10

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

Tap for more steps...

Step 2.7.2.10.1

Factor $2$ out of $2 i$ .

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

Step 2.7.2.10.2

Cancel the common factors.

Tap for more steps...

Step 2.7.2.10.2.1

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

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

Step 2.7.2.10.2.2

Cancel the common factor.

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

Step 2.7.2.10.2.3

Rewrite the expression.

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

Step 2.7.2.11

Cancel the common factor of $i$ .

Tap for more steps...

Step 2.7.2.11.1

Cancel the common factor.

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

Step 2.7.2.11.2

Rewrite the expression.

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

Step 2.7.2.11.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

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

Step 2.7.2.12

Multiply $- (- 1 \cdot 1)$ .

Tap for more steps...

Step 2.7.2.12.1

Multiply $- 1$ by $1$ .

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

Step 2.7.2.12.2

Multiply $- 1$ by $- 1$ .

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

Step 2.7.3

Add $- 2$ and $1$ .

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

Step 2.8

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.8.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x + i y, 1, 1$

Step 2.8.2

Remove parentheses.

$x + i y, 1, 1$

Step 2.8.3

The LCM of one and any expression is the expression.

$x + i y$

Step 2.9

Multiply each term in $\frac{2}{x + i y} = - 1 - 2 i$ by $x + i y$ to eliminate the fractions.

Tap for more steps...

Step 2.9.1

Multiply each term in $\frac{2}{x + i y} = - 1 - 2 i$ by $x + i y$ .

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

Step 2.9.2

Simplify the left side.

Tap for more steps...

Step 2.9.2.1

Cancel the common factor of $x + i y$ .

Tap for more steps...

Step 2.9.2.1.1

Cancel the common factor.

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

Step 2.9.2.1.2

Rewrite the expression.

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

Step 2.9.3

Simplify the right side.

Tap for more steps...

Step 2.9.3.1

Simplify each term.

Tap for more steps...

Step 2.9.3.1.1

Apply the distributive property.

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

Step 2.9.3.1.2

Apply the distributive property.

$2 = - x - i y - 2 i x - 2 i (i y)$

Step 2.9.3.1.3

Multiply $- 2 i (i y)$ .

Tap for more steps...

Step 2.9.3.1.3.1

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

$2 = - x - i y - 2 i x - 2 (i^{1} i) y$

Step 2.9.3.1.3.2

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

$2 = - x - i y - 2 i x - 2 (i^{1} i^{1}) y$

Step 2.9.3.1.3.3

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

$2 = - x - i y - 2 i x - 2 i^{1 + 1} y$

Step 2.9.3.1.3.4

Add $1$ and $1$ .

$2 = - x - i y - 2 i x - 2 i^{2} y$

Step 2.9.3.1.4

Simplify each term.

Tap for more steps...

Step 2.9.3.1.4.1

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

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

Step 2.9.3.1.4.2

Multiply $- 2$ by $- 1$ .

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

Step 2.10

Solve the equation.

Tap for more steps...

Step 2.10.1

Rewrite the equation as $- x - i y - 2 i x + 2 y = 2$ .

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

Step 2.10.2

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

Tap for more steps...

Step 2.10.2.1

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

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

Step 2.10.2.2

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

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

Step 2.10.3

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

Tap for more steps...

Step 2.10.3.1

Factor $x$ out of $- x$ .

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

Step 2.10.3.2

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

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

Step 2.10.3.3

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

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

Step 2.10.4

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

Tap for more steps...

Step 2.10.4.1

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

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

Step 2.10.4.2

Simplify the left side.

Tap for more steps...

Step 2.10.4.2.1

Cancel the common factor of $- 1 - 2 i$ .

Tap for more steps...

Step 2.10.4.2.1.1

Cancel the common factor.

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

Step 2.10.4.2.1.2

Divide $x$ by $1$ .

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

Step 2.10.4.3

Simplify the right side.

Tap for more steps...

Step 2.10.4.3.1

Simplify each term.

Tap for more steps...

Step 2.10.4.3.1.1

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

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

Step 2.10.4.3.1.2

Multiply.

Tap for more steps...

Step 2.10.4.3.1.2.1

Combine.

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

Step 2.10.4.3.1.2.2

Simplify the numerator.

Tap for more steps...

Step 2.10.4.3.1.2.2.1

Apply the distributive property.

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

Step 2.10.4.3.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.10.4.3.1.2.2.3

Multiply $2$ by $2$ .

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

Step 2.10.4.3.1.2.3

Simplify the denominator.

Tap for more steps...

Step 2.10.4.3.1.2.3.1

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

Tap for more steps...

Step 2.10.4.3.1.2.3.1.1

Apply the distributive property.

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

Step 2.10.4.3.1.2.3.1.2

Apply the distributive property.

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

Step 2.10.4.3.1.2.3.1.3

Apply the distributive property.

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

Step 2.10.4.3.1.2.3.2

Simplify.

Tap for more steps...

Step 2.10.4.3.1.2.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.4.3.1.2.3.2.2

Multiply $2$ by $- 1$ .

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

Step 2.10.4.3.1.2.3.2.3

Multiply $- 1$ by $- 2$ .

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

Step 2.10.4.3.1.2.3.2.4

Multiply $2$ by $- 2$ .

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

Step 2.10.4.3.1.2.3.2.5

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

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

Step 2.10.4.3.1.2.3.2.6

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

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

Step 2.10.4.3.1.2.3.2.7

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

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

Step 2.10.4.3.1.2.3.2.8

Add $1$ and $1$ .

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

Step 2.10.4.3.1.2.3.2.9

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

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

Step 2.10.4.3.1.2.3.2.10

Add $1$ and $0$ .

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

Step 2.10.4.3.1.2.3.3

Simplify each term.

Tap for more steps...

Step 2.10.4.3.1.2.3.3.1

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

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

Step 2.10.4.3.1.2.3.3.2

Multiply $- 4$ by $- 1$ .

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

Step 2.10.4.3.1.2.3.4

Add $1$ and $4$ .

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

Step 2.10.4.3.1.3

Split the fraction $\frac{- 2 + 4 i}{5}$ into two fractions.

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

Step 2.10.4.3.1.4

Move the negative in front of the fraction.

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

Step 2.10.4.3.1.5

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

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

Step 2.10.4.3.1.6

Multiply.

Tap for more steps...

Step 2.10.4.3.1.6.1

Combine.

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

Step 2.10.4.3.1.6.2

Simplify the numerator.

Tap for more steps...

Step 2.10.4.3.1.6.2.1

Apply the distributive property.

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

Step 2.10.4.3.1.6.2.2

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

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

Step 2.10.4.3.1.6.2.3

Multiply $i y (2 i)$ .

Tap for more steps...

Step 2.10.4.3.1.6.2.3.1

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

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

Step 2.10.4.3.1.6.2.3.2

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

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

Step 2.10.4.3.1.6.2.3.3

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

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

Step 2.10.4.3.1.6.2.3.4

Add $1$ and $1$ .

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

Step 2.10.4.3.1.6.2.4

Simplify each term.

Tap for more steps...

Step 2.10.4.3.1.6.2.4.1

Rewrite $- 1 (i y)$ as $- (i y)$ .

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

Step 2.10.4.3.1.6.2.4.2

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

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

Step 2.10.4.3.1.6.2.4.3

Multiply $2$ by $- 1$ .

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

Step 2.10.4.3.1.6.2.4.4

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

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

Step 2.10.4.3.1.6.3

Simplify the denominator.

Tap for more steps...

Step 2.10.4.3.1.6.3.1

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

Tap for more steps...

Step 2.10.4.3.1.6.3.1.1

Apply the distributive property.

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

Step 2.10.4.3.1.6.3.1.2

Apply the distributive property.

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

Step 2.10.4.3.1.6.3.1.3

Apply the distributive property.

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

Step 2.10.4.3.1.6.3.2

Simplify.

Tap for more steps...

Step 2.10.4.3.1.6.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.4.3.1.6.3.2.2

Multiply $2$ by $- 1$ .

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

Step 2.10.4.3.1.6.3.2.3

Multiply $- 1$ by $- 2$ .

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

Step 2.10.4.3.1.6.3.2.4

Multiply $2$ by $- 2$ .

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

Step 2.10.4.3.1.6.3.2.5

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

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

Step 2.10.4.3.1.6.3.2.6

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

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

Step 2.10.4.3.1.6.3.2.7

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

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

Step 2.10.4.3.1.6.3.2.8

Add $1$ and $1$ .

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

Step 2.10.4.3.1.6.3.2.9

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

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

Step 2.10.4.3.1.6.3.2.10

Add $1$ and $0$ .

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

Step 2.10.4.3.1.6.3.3

Simplify each term.

Tap for more steps...

Step 2.10.4.3.1.6.3.3.1

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

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

Step 2.10.4.3.1.6.3.3.2

Multiply $- 4$ by $- 1$ .

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

Step 2.10.4.3.1.6.3.4

Add $1$ and $4$ .

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

Step 2.10.4.3.1.7

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

Tap for more steps...

Step 2.10.4.3.1.7.1

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

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

Step 2.10.4.3.1.7.2

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

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

Step 2.10.4.3.1.7.3

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

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

Step 2.10.4.3.1.8

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

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

Step 2.10.4.3.1.9

Multiply.

Tap for more steps...

Step 2.10.4.3.1.9.1

Combine.

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

Step 2.10.4.3.1.9.2

Simplify the numerator.

Tap for more steps...

Step 2.10.4.3.1.9.2.1

Apply the distributive property.

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

Step 2.10.4.3.1.9.2.2

Multiply $- 1$ by $- 2$ .

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

Step 2.10.4.3.1.9.2.3

Multiply $2$ by $- 2$ .

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

Step 2.10.4.3.1.9.3

Simplify the denominator.

Tap for more steps...

Step 2.10.4.3.1.9.3.1

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

Tap for more steps...

Step 2.10.4.3.1.9.3.1.1

Apply the distributive property.

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

Step 2.10.4.3.1.9.3.1.2

Apply the distributive property.

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

Step 2.10.4.3.1.9.3.1.3

Apply the distributive property.

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

Step 2.10.4.3.1.9.3.2

Simplify.

Tap for more steps...

Step 2.10.4.3.1.9.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.4.3.1.9.3.2.2

Multiply $2$ by $- 1$ .

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

Step 2.10.4.3.1.9.3.2.3

Multiply $- 1$ by $- 2$ .

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

Step 2.10.4.3.1.9.3.2.4

Multiply $2$ by $- 2$ .

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

Step 2.10.4.3.1.9.3.2.5

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

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

Step 2.10.4.3.1.9.3.2.6

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

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

Step 2.10.4.3.1.9.3.2.7

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

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

Step 2.10.4.3.1.9.3.2.8

Add $1$ and $1$ .

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

Step 2.10.4.3.1.9.3.2.9

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

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

Step 2.10.4.3.1.9.3.2.10

Add $1$ and $0$ .

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

Step 2.10.4.3.1.9.3.3

Simplify each term.

Tap for more steps...

Step 2.10.4.3.1.9.3.3.1

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

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

Step 2.10.4.3.1.9.3.3.2

Multiply $- 4$ by $- 1$ .

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

Step 2.10.4.3.1.9.3.4

Add $1$ and $4$ .

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

Step 2.10.4.3.1.10

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

Tap for more steps...

Step 2.10.4.3.1.10.1

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

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

Step 2.10.4.3.1.10.2

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

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

Step 2.10.4.3.1.10.3

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

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

Step 2.10.4.3.2

Combine the numerators over the common denominator.

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

Step 2.10.4.3.3

Simplify each term.

Tap for more steps...

Step 2.10.4.3.3.1

Apply the distributive property.

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

Step 2.10.4.3.3.2

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

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

Step 2.10.4.3.3.3

Apply the distributive property.

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

Step 2.10.4.3.3.4

Multiply $2$ by $1$ .

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

Step 2.10.4.3.3.5

Multiply $- 2$ by $2$ .

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

Step 2.10.4.3.4

Combine the opposite terms in $- 2 + 4 i + y (- i) - 2 y + 2 y - 4 y i$ .

Tap for more steps...

Step 2.10.4.3.4.1

Add $- 2 y$ and $2 y$ .

$x = \frac{- 2 + 4 i + y (- i) + 0 - 4 y i}{5}$

Step 2.10.4.3.4.2

Add $- 2 + 4 i + y (- i)$ and $0$ .

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

Step 2.10.4.3.5

Subtract $4 y i$ from $y (- i)$ .

Tap for more steps...

Step 2.10.4.3.5.1

Reorder $y$ and $- 1$ .

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

Step 2.10.4.3.5.2

Subtract $4 y i$ from $- 1 \cdot y i$ .

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

Step 2.10.4.3.6

Reorder terms.

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

Step 2.10.4.3.7

Factor $- 1$ out of $- 5 i y$ .

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

Step 2.10.4.3.8

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

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

Step 2.10.4.3.9

Factor $- 1$ out of $- (5 i y) - 1 (2)$ .

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

Step 2.10.4.3.10

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

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

Step 2.10.4.3.11

Factor $- 1$ out of $- (5 i y + 2) - (- 4 i)$ .

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

Step 2.10.4.3.12

Simplify the expression.

Tap for more steps...

Step 2.10.4.3.12.1

Rewrite $- (5 i y + 2 - 4 i)$ as $- 1 (5 i y + 2 - 4 i)$ .

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

Step 2.10.4.3.12.2

Move the negative in front of the fraction.

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

Step 2.11

The complete solution is the result of both the positive and negative portions of the solution.

$x = - \frac{13 i y - 6 + 4 i}{13}$

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

$x = - \frac{13 i y - 6 + 4 i}{13}$

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