Algebra Examples

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

Step 1

Multiply both sides by $4 - 5 i$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{(5 - 3 i) (x + i y)}{4 - 5 i} (4 - 5 i)$ .

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Step 2.1.1.1

Cancel the common factor of $4 - 5 i$ .

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Step 2.1.1.1.1

Cancel the common factor.

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

Step 2.1.1.1.2

Rewrite the expression.

$(5 - 3 i) (x + i y) = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.1.1.2

Expand $(5 - 3 i) (x + i y)$ using the FOIL Method.

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Step 2.1.1.2.1

Apply the distributive property.

$5 (x + i y) - 3 i (x + i y) = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.1.1.2.2

Apply the distributive property.

$5 x + 5 (i y) - 3 i (x + i y) = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.1.1.2.3

Apply the distributive property.

$5 x + 5 (i y) - 3 i x - 3 i (i y) = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.1.1.3

Simplify terms.

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Step 2.1.1.3.1

Simplify each term.

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Step 2.1.1.3.1.1

Multiply $- 3 i (i y)$ .

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Step 2.1.1.3.1.1.1

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

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

Step 2.1.1.3.1.1.2

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

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

Step 2.1.1.3.1.1.3

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

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

Step 2.1.1.3.1.1.4

Add $1$ and $1$ .

$5 x + 5 i y - 3 i x - 3 i^{2} y = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.1.1.3.1.2

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

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

Step 2.1.1.3.1.3

Multiply $- 3$ by $- 1$ .

$5 x + 5 i y - 3 i x + 3 y = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.1.1.3.2

Reorder.

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Step 2.1.1.3.2.1

Move $i$ .

$5 x + 5 y i - 3 i x + 3 y = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.1.1.3.2.2

Move $i$ .

$5 x + 5 y i - 3 x i + 3 y = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.1.1.3.2.3

Move $5 y i$ .

$5 x - 3 x i + 5 y i + 3 y = ({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$ .

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Step 2.2.1.1

Expand $({(2 + i)}^{2} - {(3 - 4 i)}^{2}) (4 - 5 i)$ using the FOIL Method.

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Step 2.2.1.1.1

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = {(2 + i)}^{2} (4 - 5 i) - {(3 - 4 i)}^{2} (4 - 5 i)$

Step 2.2.1.1.2

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = {(2 + i)}^{2} \cdot 4 + {(2 + i)}^{2} (- 5 i) - {(3 - 4 i)}^{2} (4 - 5 i)$

Step 2.2.1.1.3

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = {(2 + i)}^{2} \cdot 4 + {(2 + i)}^{2} (- 5 i) - {(3 - 4 i)}^{2} \cdot 4 - {(3 - 4 i)}^{2} (- 5 i)$

Step 2.2.1.2

Simplify terms.

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Step 2.2.1.2.1

Simplify each term.

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Step 2.2.1.2.1.1

Move $4$ to the left of ${(2 + i)}^{2}$ .

$5 x - 3 x i + 5 y i + 3 y = 4 \cdot {(2 + i)}^{2} + {(2 + i)}^{2} (- 5 i) - {(3 - 4 i)}^{2} \cdot 4 - {(3 - 4 i)}^{2} (- 5 i)$

Step 2.2.1.2.1.2

Multiply $4$ by $- 1$ .

$5 x - 3 x i + 5 y i + 3 y = 4 {(2 + i)}^{2} + {(2 + i)}^{2} (- 5 i) - 4 {(3 - 4 i)}^{2} - {(3 - 4 i)}^{2} (- 5 i)$

Step 2.2.1.2.1.3

Multiply $- 5$ by $- 1$ .

$5 x - 3 x i + 5 y i + 3 y = 4 {(2 + i)}^{2} + {(2 + i)}^{2} (- 5 i) - 4 {(3 - 4 i)}^{2} + 5 {(3 - 4 i)}^{2} i$

Step 2.2.1.2.2

Reorder factors in $4 {(2 + i)}^{2} + {(2 + i)}^{2} (- 5 i) - 4 {(3 - 4 i)}^{2} + 5 {(3 - 4 i)}^{2} i$ .

$5 x - 3 x i + 5 y i + 3 y = 4 {(2 + i)}^{2} - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $4 {(2 + i)}^{2} - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

$5 x - 3 x i + 5 y i + 3 y = 4 ((2 + i) (2 + i)) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.2

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

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Step 3.1.1.2.1

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 4 (2 (2 + i) + i (2 + i)) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.2.2

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 4 (2 \cdot 2 + 2 i + i (2 + i)) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.2.3

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 4 (2 \cdot 2 + 2 i + i \cdot 2 + i i) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.3

Simplify and combine like terms.

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Step 3.1.1.3.1

Simplify each term.

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Step 3.1.1.3.1.1

Multiply $2$ by $2$ .

$5 x - 3 x i + 5 y i + 3 y = 4 (4 + 2 i + i \cdot 2 + i i) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.3.1.2

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

$5 x - 3 x i + 5 y i + 3 y = 4 (4 + 2 i + 2 \cdot i + i i) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.3.1.3

Multiply $i i$ .

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Step 3.1.1.3.1.3.1

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

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

Step 3.1.1.3.1.3.2

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

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

Step 3.1.1.3.1.3.3

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

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

Step 3.1.1.3.1.3.4

Add $1$ and $1$ .

$5 x - 3 x i + 5 y i + 3 y = 4 (4 + 2 i + 2 i + i^{2}) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.3.1.4

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

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

Step 3.1.1.3.2

Subtract $1$ from $4$ .

$5 x - 3 x i + 5 y i + 3 y = 4 (3 + 2 i + 2 i) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.3.3

Add $2 i$ and $2 i$ .

$5 x - 3 x i + 5 y i + 3 y = 4 (3 + 4 i) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.4

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 4 \cdot 3 + 4 (4 i) - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.5

Multiply $4$ by $3$ .

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

Step 3.1.1.6

Multiply $4$ by $4$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i {(2 + i)}^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.7

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i ((2 + i) (2 + i)) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.8

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

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Step 3.1.1.8.1

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (2 (2 + i) + i (2 + i)) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.8.2

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (2 \cdot 2 + 2 i + i (2 + i)) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.8.3

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (2 \cdot 2 + 2 i + i \cdot 2 + i i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9

Simplify and combine like terms.

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Step 3.1.1.9.1

Simplify each term.

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Step 3.1.1.9.1.1

Multiply $2$ by $2$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (4 + 2 i + i \cdot 2 + i i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9.1.2

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (4 + 2 i + 2 \cdot i + i i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9.1.3

Multiply $i i$ .

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Step 3.1.1.9.1.3.1

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (4 + 2 i + 2 i + i^{1} i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9.1.3.2

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (4 + 2 i + 2 i + i^{1} i^{1}) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9.1.3.3

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (4 + 2 i + 2 i + i^{1 + 1}) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9.1.3.4

Add $1$ and $1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (4 + 2 i + 2 i + i^{2}) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9.1.4

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (4 + 2 i + 2 i - 1) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9.2

Subtract $1$ from $4$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (3 + 2 i + 2 i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.9.3

Add $2 i$ and $2 i$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i (3 + 4 i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.10

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 5 i \cdot 3 - 5 i (4 i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.11

Multiply $3$ by $- 5$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i - 5 i (4 i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.12

Multiply $- 5 i (4 i)$ .

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Step 3.1.1.12.1

Multiply $4$ by $- 5$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i - 20 i i - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.12.2

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i - 20 (i^{1} i) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.12.3

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i - 20 (i^{1} i^{1}) - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.12.4

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i - 20 i^{1 + 1} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.12.5

Add $1$ and $1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i - 20 i^{2} - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.13

Simplify each term.

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Step 3.1.1.13.1

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i - 20 \cdot - 1 - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.13.2

Multiply $- 20$ by $- 1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 {(3 - 4 i)}^{2} + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.14

Rewrite ${(3 - 4 i)}^{2}$ as $(3 - 4 i) (3 - 4 i)$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 ((3 - 4 i) (3 - 4 i)) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.15

Expand $(3 - 4 i) (3 - 4 i)$ using the FOIL Method.

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Step 3.1.1.15.1

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (3 (3 - 4 i) - 4 i (3 - 4 i)) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.15.2

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (3 \cdot 3 + 3 (- 4 i) - 4 i (3 - 4 i)) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.15.3

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (3 \cdot 3 + 3 (- 4 i) - 4 i \cdot 3 - 4 i (- 4 i)) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16

Simplify and combine like terms.

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Step 3.1.1.16.1

Simplify each term.

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Step 3.1.1.16.1.1

Multiply $3$ by $3$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 + 3 (- 4 i) - 4 i \cdot 3 - 4 i (- 4 i)) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.2

Multiply $- 4$ by $3$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 4 i \cdot 3 - 4 i (- 4 i)) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.3

Multiply $3$ by $- 4$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 12 i - 4 i (- 4 i)) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.4

Multiply $- 4 i (- 4 i)$ .

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Step 3.1.1.16.1.4.1

Multiply $- 4$ by $- 4$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 12 i + 16 i i) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.4.2

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 12 i + 16 (i^{1} i)) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.4.3

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 12 i + 16 (i^{1} i^{1})) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.4.4

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 12 i + 16 i^{1 + 1}) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.4.5

Add $1$ and $1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 12 i + 16 i^{2}) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.5

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 12 i + 16 \cdot - 1) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.1.6

Multiply $16$ by $- 1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (9 - 12 i - 12 i - 16) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.2

Subtract $16$ from $9$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (- 7 - 12 i - 12 i) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.16.3

Subtract $12 i$ from $- 12 i$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 (- 7 - 24 i) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.17

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 - 4 \cdot - 7 - 4 (- 24 i) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.18

Multiply $- 4$ by $- 7$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 - 4 (- 24 i) + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.19

Multiply $- 24$ by $- 4$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i {(3 - 4 i)}^{2}$

Step 3.1.1.20

Rewrite ${(3 - 4 i)}^{2}$ as $(3 - 4 i) (3 - 4 i)$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i ((3 - 4 i) (3 - 4 i))$

Step 3.1.1.21

Expand $(3 - 4 i) (3 - 4 i)$ using the FOIL Method.

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Step 3.1.1.21.1

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (3 (3 - 4 i) - 4 i (3 - 4 i))$

Step 3.1.1.21.2

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (3 \cdot 3 + 3 (- 4 i) - 4 i (3 - 4 i))$

Step 3.1.1.21.3

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (3 \cdot 3 + 3 (- 4 i) - 4 i \cdot 3 - 4 i (- 4 i))$

Step 3.1.1.22

Simplify and combine like terms.

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Step 3.1.1.22.1

Simplify each term.

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Step 3.1.1.22.1.1

Multiply $3$ by $3$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 + 3 (- 4 i) - 4 i \cdot 3 - 4 i (- 4 i))$

Step 3.1.1.22.1.2

Multiply $- 4$ by $3$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 4 i \cdot 3 - 4 i (- 4 i))$

Step 3.1.1.22.1.3

Multiply $3$ by $- 4$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 12 i - 4 i (- 4 i))$

Step 3.1.1.22.1.4

Multiply $- 4 i (- 4 i)$ .

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Step 3.1.1.22.1.4.1

Multiply $- 4$ by $- 4$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 12 i + 16 i i)$

Step 3.1.1.22.1.4.2

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 12 i + 16 (i^{1} i))$

Step 3.1.1.22.1.4.3

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 12 i + 16 (i^{1} i^{1}))$

Step 3.1.1.22.1.4.4

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 12 i + 16 i^{1 + 1})$

Step 3.1.1.22.1.4.5

Add $1$ and $1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 12 i + 16 i^{2})$

Step 3.1.1.22.1.5

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 12 i + 16 \cdot - 1)$

Step 3.1.1.22.1.6

Multiply $16$ by $- 1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (9 - 12 i - 12 i - 16)$

Step 3.1.1.22.2

Subtract $16$ from $9$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (- 7 - 12 i - 12 i)$

Step 3.1.1.22.3

Subtract $12 i$ from $- 12 i$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i (- 7 - 24 i)$

Step 3.1.1.23

Apply the distributive property.

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i + 5 i \cdot - 7 + 5 i (- 24 i)$

Step 3.1.1.24

Multiply $- 7$ by $5$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i - 35 i + 5 i (- 24 i)$

Step 3.1.1.25

Multiply $5 i (- 24 i)$ .

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Step 3.1.1.25.1

Multiply $- 24$ by $5$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i - 35 i - 120 i i$

Step 3.1.1.25.2

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i - 35 i - 120 (i^{1} i)$

Step 3.1.1.25.3

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i - 35 i - 120 (i^{1} i^{1})$

Step 3.1.1.25.4

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i - 35 i - 120 i^{1 + 1}$

Step 3.1.1.25.5

Add $1$ and $1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i - 35 i - 120 i^{2}$

Step 3.1.1.26

Simplify each term.

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Step 3.1.1.26.1

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

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i - 35 i - 120 \cdot - 1$

Step 3.1.1.26.2

Multiply $- 120$ by $- 1$ .

$5 x - 3 x i + 5 y i + 3 y = 12 + 16 i - 15 i + 20 + 28 + 96 i - 35 i + 120$

Step 3.1.2

Simplify by adding terms.

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Step 3.1.2.1

Add $12$ and $20$ .

$5 x - 3 x i + 5 y i + 3 y = 32 + 16 i - 15 i + 28 + 96 i - 35 i + 120$

Step 3.1.2.2

Simplify by adding numbers.

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Step 3.1.2.2.1

Add $32$ and $28$ .

$5 x - 3 x i + 5 y i + 3 y = 60 + 16 i - 15 i + 96 i - 35 i + 120$

Step 3.1.2.2.2

Add $60$ and $120$ .

$5 x - 3 x i + 5 y i + 3 y = 180 + 16 i - 15 i + 96 i - 35 i$

Step 3.1.2.3

Subtract $15 i$ from $16 i$ .

$5 x - 3 x i + 5 y i + 3 y = 180 + i + 96 i - 35 i$

Step 3.1.2.4

Add $i$ and $96 i$ .

$5 x - 3 x i + 5 y i + 3 y = 180 + 97 i - 35 i$

Step 3.1.2.5

Subtract $35 i$ from $97 i$ .

$5 x - 3 x i + 5 y i + 3 y = 180 + 62 i$

Step 3.2

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

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Step 3.2.1

Subtract $5 y i$ from both sides of the equation.

$5 x - 3 x i + 3 y = 180 + 62 i - 5 y i$

Step 3.2.2

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

$5 x - 3 x i = 180 + 62 i - 5 y i - 3 y$

Step 3.3

Factor $x$ out of $5 x - 3 x i$ .

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Step 3.3.1

Factor $x$ out of $5 x$ .

$x \cdot 5 - 3 x i = 180 + 62 i - 5 y i - 3 y$

Step 3.3.2

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

$x \cdot 5 + x (- 3 i) = 180 + 62 i - 5 y i - 3 y$

Step 3.3.3

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

$x (5 - 3 i) = 180 + 62 i - 5 y i - 3 y$

Step 3.4

Divide each term in $x (5 - 3 i) = 180 + 62 i - 5 y i - 3 y$ by $5 - 3 i$ and simplify.

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Step 3.4.1

Divide each term in $x (5 - 3 i) = 180 + 62 i - 5 y i - 3 y$ by $5 - 3 i$ .

$\frac{x (5 - 3 i)}{5 - 3 i} = \frac{180}{5 - 3 i} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $5 - 3 i$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{x (5 - 3 i)}{5 - 3 i} = \frac{180}{5 - 3 i} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{180}{5 - 3 i} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Multiply the numerator and denominator of $\frac{180}{5 - 3 i}$ by the conjugate of $5 - 3 i$ to make the denominator real.

$x = \frac{180}{5 - 3 i} \cdot \frac{5 + 3 i}{5 + 3 i} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2

Multiply.

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Step 3.4.3.1.2.1

Combine.

$x = \frac{180 (5 + 3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.2

Simplify the numerator.

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Step 3.4.3.1.2.2.1

Apply the distributive property.

$x = \frac{180 \cdot 5 + 180 (3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.2.2

Multiply $180$ by $5$ .

$x = \frac{900 + 180 (3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.2.3

Multiply $3$ by $180$ .

$x = \frac{900 + 540 i}{(5 - 3 i) (5 + 3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3

Simplify the denominator.

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Step 3.4.3.1.2.3.1

Expand $(5 - 3 i) (5 + 3 i)$ using the FOIL Method.

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Step 3.4.3.1.2.3.1.1

Apply the distributive property.

$x = \frac{900 + 540 i}{5 (5 + 3 i) - 3 i (5 + 3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.1.2

Apply the distributive property.

$x = \frac{900 + 540 i}{5 \cdot 5 + 5 (3 i) - 3 i (5 + 3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.1.3

Apply the distributive property.

$x = \frac{900 + 540 i}{5 \cdot 5 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2

Simplify.

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Step 3.4.3.1.2.3.2.1

Multiply $5$ by $5$ .

$x = \frac{900 + 540 i}{25 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.2

Multiply $3$ by $5$ .

$x = \frac{900 + 540 i}{25 + 15 i - 3 i \cdot 5 - 3 i (3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.3

Multiply $5$ by $- 3$ .

$x = \frac{900 + 540 i}{25 + 15 i - 15 i - 3 i (3 i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.4

Multiply $3$ by $- 3$ .

$x = \frac{900 + 540 i}{25 + 15 i - 15 i - 9 i i} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.5

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

$x = \frac{900 + 540 i}{25 + 15 i - 15 i - 9 (i^{1} i)} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.6

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

$x = \frac{900 + 540 i}{25 + 15 i - 15 i - 9 (i^{1} i^{1})} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.7

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

$x = \frac{900 + 540 i}{25 + 15 i - 15 i - 9 i^{1 + 1}} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.8

Add $1$ and $1$ .

$x = \frac{900 + 540 i}{25 + 15 i - 15 i - 9 i^{2}} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.9

Subtract $15 i$ from $15 i$ .

$x = \frac{900 + 540 i}{25 + 0 - 9 i^{2}} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.2.10

Add $25$ and $0$ .

$x = \frac{900 + 540 i}{25 - 9 i^{2}} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.3

Simplify each term.

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Step 3.4.3.1.2.3.3.1

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

$x = \frac{900 + 540 i}{25 - 9 \cdot - 1} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.3.2

Multiply $- 9$ by $- 1$ .

$x = \frac{900 + 540 i}{25 + 9} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.2.3.4

Add $25$ and $9$ .

$x = \frac{900 + 540 i}{34} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.3

Cancel the common factor of $900 + 540 i$ and $34$ .

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Step 3.4.3.1.3.1

Factor $2$ out of $900$ .

$x = \frac{2 (450) + 540 i}{34} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.3.2

Factor $2$ out of $540 i$ .

$x = \frac{2 (450) + 2 (270 i)}{34} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.3.3

Factor $2$ out of $2 (450) + 2 (270 i)$ .

$x = \frac{2 (450 + 270 i)}{34} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.3.4

Cancel the common factors.

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Step 3.4.3.1.3.4.1

Factor $2$ out of $34$ .

$x = \frac{2 (450 + 270 i)}{2 \cdot 17} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.3.4.2

Cancel the common factor.

$x = \frac{2 (450 + 270 i)}{2 \cdot 17} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.3.4.3

Rewrite the expression.

$x = \frac{450 + 270 i}{17} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.4

Split the fraction $\frac{450 + 270 i}{17}$ into two fractions.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{62 i}{5 - 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.5

Multiply the numerator and denominator of $\frac{62 i}{5 - 3 i}$ by the conjugate of $5 - 3 i$ to make the denominator real.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{62 i}{5 - 3 i} \cdot \frac{5 + 3 i}{5 + 3 i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6

Multiply.

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Step 3.4.3.1.6.1

Combine.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{62 i (5 + 3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2

Simplify the numerator.

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Step 3.4.3.1.6.2.1

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{62 i \cdot 5 + 62 i (3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.2

Multiply $5$ by $62$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{310 i + 62 i (3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.3

Multiply $62 i (3 i)$ .

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Step 3.4.3.1.6.2.3.1

Multiply $3$ by $62$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{310 i + 186 i i}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.3.2

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

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{310 i + 186 (i^{1} i)}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.3.3

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

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{310 i + 186 (i^{1} i^{1})}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.3.4

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

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{310 i + 186 i^{1 + 1}}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.3.5

Add $1$ and $1$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{310 i + 186 i^{2}}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.4

Simplify each term.

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Step 3.4.3.1.6.2.4.1

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

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{310 i + 186 \cdot - 1}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.4.2

Multiply $186$ by $- 1$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{310 i - 186}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.2.5

Reorder $310 i$ and $- 186$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{(5 - 3 i) (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3

Simplify the denominator.

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Step 3.4.3.1.6.3.1

Expand $(5 - 3 i) (5 + 3 i)$ using the FOIL Method.

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Step 3.4.3.1.6.3.1.1

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{5 (5 + 3 i) - 3 i (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.1.2

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{5 \cdot 5 + 5 (3 i) - 3 i (5 + 3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.1.3

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{5 \cdot 5 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2

Simplify.

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Step 3.4.3.1.6.3.2.1

Multiply $5$ by $5$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.2

Multiply $3$ by $5$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 15 i - 3 i \cdot 5 - 3 i (3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.3

Multiply $5$ by $- 3$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 15 i - 15 i - 3 i (3 i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.4

Multiply $3$ by $- 3$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 15 i - 15 i - 9 i i} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.5

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

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 15 i - 15 i - 9 (i^{1} i)} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.6

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

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 15 i - 15 i - 9 (i^{1} i^{1})} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.7

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

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 15 i - 15 i - 9 i^{1 + 1}} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.8

Add $1$ and $1$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 15 i - 15 i - 9 i^{2}} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.9

Subtract $15 i$ from $15 i$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 0 - 9 i^{2}} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.2.10

Add $25$ and $0$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 - 9 i^{2}} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.3

Simplify each term.

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Step 3.4.3.1.6.3.3.1

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

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 - 9 \cdot - 1} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.3.2

Multiply $- 9$ by $- 1$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{25 + 9} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.6.3.4

Add $25$ and $9$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 186 + 310 i}{34} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.7

Cancel the common factor of $- 186 + 310 i$ and $34$ .

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Step 3.4.3.1.7.1

Factor $2$ out of $- 186$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{2 (- 93) + 310 i}{34} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.7.2

Factor $2$ out of $310 i$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{2 (- 93) + 2 (155 i)}{34} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.7.3

Factor $2$ out of $2 (- 93) + 2 (155 i)$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{2 (- 93 + 155 i)}{34} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.7.4

Cancel the common factors.

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Step 3.4.3.1.7.4.1

Factor $2$ out of $34$ .

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{2 (- 93 + 155 i)}{2 \cdot 17} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.7.4.2

Cancel the common factor.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{2 (- 93 + 155 i)}{2 \cdot 17} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.7.4.3

Rewrite the expression.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 93 + 155 i}{17} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.8

Split the fraction $\frac{- 93 + 155 i}{17}$ into two fractions.

$x = \frac{450}{17} + \frac{270 i}{17} + \frac{- 93}{17} + \frac{155 i}{17} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.9

Move the negative in front of the fraction.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 5 y i}{5 - 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.10

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 5 y i}{5 - 3 i} \cdot \frac{5 + 3 i}{5 + 3 i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11

Multiply.

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Step 3.4.3.1.11.1

Combine.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 5 y i (5 + 3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2

Simplify the numerator.

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Step 3.4.3.1.11.2.1

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 5 y i \cdot 5 - 5 y i (3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2.2

Multiply $5$ by $- 5$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i - 5 y i (3 i)}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2.3

Multiply $- 5 y i (3 i)$ .

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Step 3.4.3.1.11.2.3.1

Multiply $3$ by $- 5$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i - 15 y i i}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2.3.2

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i - 15 y (i^{1} i)}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2.3.3

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i - 15 y (i^{1} i^{1})}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2.3.4

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i - 15 y i^{1 + 1}}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2.3.5

Add $1$ and $1$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i - 15 y i^{2}}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2.4

Simplify each term.

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Step 3.4.3.1.11.2.4.1

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i - 15 y \cdot - 1}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.2.4.2

Multiply $- 1$ by $- 15$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{(5 - 3 i) (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3

Simplify the denominator.

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Step 3.4.3.1.11.3.1

Expand $(5 - 3 i) (5 + 3 i)$ using the FOIL Method.

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Step 3.4.3.1.11.3.1.1

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{5 (5 + 3 i) - 3 i (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.1.2

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{5 \cdot 5 + 5 (3 i) - 3 i (5 + 3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.1.3

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{5 \cdot 5 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2

Simplify.

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Step 3.4.3.1.11.3.2.1

Multiply $5$ by $5$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.2

Multiply $3$ by $5$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 15 i - 3 i \cdot 5 - 3 i (3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.3

Multiply $5$ by $- 3$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 15 i - 15 i - 3 i (3 i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.4

Multiply $3$ by $- 3$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 15 i - 15 i - 9 i i} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.5

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 15 i - 15 i - 9 (i^{1} i)} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.6

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 15 i - 15 i - 9 (i^{1} i^{1})} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.7

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 15 i - 15 i - 9 i^{1 + 1}} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.8

Add $1$ and $1$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 15 i - 15 i - 9 i^{2}} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.9

Subtract $15 i$ from $15 i$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 0 - 9 i^{2}} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.2.10

Add $25$ and $0$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 - 9 i^{2}} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.3

Simplify each term.

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Step 3.4.3.1.11.3.3.1

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 - 9 \cdot - 1} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.3.2

Multiply $- 9$ by $- 1$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{25 + 9} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.11.3.4

Add $25$ and $9$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{- 25 y i + 15 y}{34} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.12

Factor $5 y$ out of $- 25 y i + 15 y$ .

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Step 3.4.3.1.12.1

Factor $5 y$ out of $- 25 y i$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i) + 15 y}{34} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.12.2

Factor $5 y$ out of $15 y$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i) + 5 y (3)}{34} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.12.3

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 3 y}{5 - 3 i}$

Step 3.4.3.1.13

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 3 y}{5 - 3 i} \cdot \frac{5 + 3 i}{5 + 3 i}$

Step 3.4.3.1.14

Multiply.

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Step 3.4.3.1.14.1

Combine.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 3 y (5 + 3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 3.4.3.1.14.2

Simplify the numerator.

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Step 3.4.3.1.14.2.1

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 3 y \cdot 5 - 3 y (3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 3.4.3.1.14.2.2

Multiply $5$ by $- 3$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 3 y (3 i)}{(5 - 3 i) (5 + 3 i)}$

Step 3.4.3.1.14.2.3

Multiply $3$ by $- 3$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{(5 - 3 i) (5 + 3 i)}$

Step 3.4.3.1.14.3

Simplify the denominator.

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Step 3.4.3.1.14.3.1

Expand $(5 - 3 i) (5 + 3 i)$ using the FOIL Method.

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Step 3.4.3.1.14.3.1.1

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{5 (5 + 3 i) - 3 i (5 + 3 i)}$

Step 3.4.3.1.14.3.1.2

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{5 \cdot 5 + 5 (3 i) - 3 i (5 + 3 i)}$

Step 3.4.3.1.14.3.1.3

Apply the distributive property.

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{5 \cdot 5 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)}$

Step 3.4.3.1.14.3.2

Simplify.

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Step 3.4.3.1.14.3.2.1

Multiply $5$ by $5$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 5 (3 i) - 3 i \cdot 5 - 3 i (3 i)}$

Step 3.4.3.1.14.3.2.2

Multiply $3$ by $5$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 15 i - 3 i \cdot 5 - 3 i (3 i)}$

Step 3.4.3.1.14.3.2.3

Multiply $5$ by $- 3$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 15 i - 15 i - 3 i (3 i)}$

Step 3.4.3.1.14.3.2.4

Multiply $3$ by $- 3$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 15 i - 15 i - 9 i i}$

Step 3.4.3.1.14.3.2.5

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 15 i - 15 i - 9 (i^{1} i)}$

Step 3.4.3.1.14.3.2.6

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 15 i - 15 i - 9 (i^{1} i^{1})}$

Step 3.4.3.1.14.3.2.7

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 15 i - 15 i - 9 i^{1 + 1}}$

Step 3.4.3.1.14.3.2.8

Add $1$ and $1$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 15 i - 15 i - 9 i^{2}}$

Step 3.4.3.1.14.3.2.9

Subtract $15 i$ from $15 i$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 0 - 9 i^{2}}$

Step 3.4.3.1.14.3.2.10

Add $25$ and $0$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 - 9 i^{2}}$

Step 3.4.3.1.14.3.3

Simplify each term.

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Step 3.4.3.1.14.3.3.1

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 - 9 \cdot - 1}$

Step 3.4.3.1.14.3.3.2

Multiply $- 9$ by $- 1$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{25 + 9}$

Step 3.4.3.1.14.3.4

Add $25$ and $9$ .

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{- 15 y - 9 y i}{34}$

Step 3.4.3.1.15

Factor $3 y$ out of $- 15 y - 9 y i$ .

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Step 3.4.3.1.15.1

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{3 y (- 5) - 9 y i}{34}$

Step 3.4.3.1.15.2

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{3 y (- 5) + 3 y (- 3 i)}{34}$

Step 3.4.3.1.15.3

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

$x = \frac{450}{17} + \frac{270 i}{17} - \frac{93}{17} + \frac{155 i}{17} + \frac{5 y (- 5 i + 3)}{34} + \frac{3 y (- 5 - 3 i)}{34}$

Step 3.4.3.2

Simplify terms.

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Step 3.4.3.2.1

Combine the numerators over the common denominator.

$x = \frac{450 + 270 i - 93 + 155 i}{17} + \frac{5 y (- 5 i + 3) + 3 y (- 5 - 3 i)}{34}$

Step 3.4.3.2.2

Subtract $93$ from $450$ .

$x = \frac{357 + 270 i + 155 i}{17} + \frac{5 y (- 5 i + 3) + 3 y (- 5 - 3 i)}{34}$

Step 3.4.3.2.3

Add $270 i$ and $155 i$ .

$x = \frac{357 + 425 i}{17} + \frac{5 y (- 5 i + 3) + 3 y (- 5 - 3 i)}{34}$

Step 3.4.3.3

Simplify each term.

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Step 3.4.3.3.1

Apply the distributive property.

$x = \frac{357 + 425 i}{17} + \frac{5 y (- 5 i) + 5 y \cdot 3 + 3 y (- 5 - 3 i)}{34}$

Step 3.4.3.3.2

Multiply $- 5$ by $5$ .

$x = \frac{357 + 425 i}{17} + \frac{- 25 y i + 5 y \cdot 3 + 3 y (- 5 - 3 i)}{34}$

Step 3.4.3.3.3

Multiply $3$ by $5$ .

$x = \frac{357 + 425 i}{17} + \frac{- 25 y i + 15 y + 3 y (- 5 - 3 i)}{34}$

Step 3.4.3.3.4

Apply the distributive property.

$x = \frac{357 + 425 i}{17} + \frac{- 25 y i + 15 y + 3 y \cdot - 5 + 3 y (- 3 i)}{34}$

Step 3.4.3.3.5

Multiply $- 5$ by $3$ .

$x = \frac{357 + 425 i}{17} + \frac{- 25 y i + 15 y - 15 y + 3 y (- 3 i)}{34}$

Step 3.4.3.3.6

Multiply $- 3$ by $3$ .

$x = \frac{357 + 425 i}{17} + \frac{- 25 y i + 15 y - 15 y - 9 y i}{34}$

Step 3.4.3.4

Simplify by adding terms.

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Step 3.4.3.4.1

Combine the opposite terms in $- 25 y i + 15 y - 15 y - 9 y i$ .

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Step 3.4.3.4.1.1

Subtract $15 y$ from $15 y$ .

$x = \frac{357 + 425 i}{17} + \frac{- 25 y i + 0 - 9 y i}{34}$

Step 3.4.3.4.1.2

Add $- 25 y i$ and $0$ .

$x = \frac{357 + 425 i}{17} + \frac{- 25 y i - 9 y i}{34}$

Step 3.4.3.4.2

Subtract $9 y i$ from $- 25 y i$ .

$x = \frac{357 + 425 i}{17} + \frac{- 34 y i}{34}$

Step 3.4.3.5

Simplify each term.

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Step 3.4.3.5.1

Cancel the common factor of $357 + 425 i$ and $17$ .

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Step 3.4.3.5.1.1

Factor $17$ out of $357$ .

$x = \frac{17 \cdot 21 + 425 i}{17} + \frac{- 34 y i}{34}$

Step 3.4.3.5.1.2

Factor $17$ out of $425 i$ .

$x = \frac{17 \cdot 21 + 17 (25 i)}{17} + \frac{- 34 y i}{34}$

Step 3.4.3.5.1.3

Factor $17$ out of $17 (21) + 17 (25 i)$ .

$x = \frac{17 (21 + 25 i)}{17} + \frac{- 34 y i}{34}$

Step 3.4.3.5.1.4

Cancel the common factors.

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Step 3.4.3.5.1.4.1

Factor $17$ out of $17$ .

$x = \frac{17 (21 + 25 i)}{17 (1)} + \frac{- 34 y i}{34}$

Step 3.4.3.5.1.4.2

Cancel the common factor.

$x = \frac{17 (21 + 25 i)}{17 \cdot 1} + \frac{- 34 y i}{34}$

Step 3.4.3.5.1.4.3

Rewrite the expression.

$x = \frac{21 + 25 i}{1} + \frac{- 34 y i}{34}$

Step 3.4.3.5.1.4.4

Divide $21 + 25 i$ by $1$ .

$x = 21 + 25 i + \frac{- 34 y i}{34}$

Step 3.4.3.5.2

Cancel the common factor of $- 34$ and $34$ .

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Step 3.4.3.5.2.1

Factor $34$ out of $- 34 y i$ .

$x = 21 + 25 i + \frac{34 (- y i)}{34}$

Step 3.4.3.5.2.2

Cancel the common factors.

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Step 3.4.3.5.2.2.1

Factor $34$ out of $34$ .

$x = 21 + 25 i + \frac{34 (- y i)}{34 (1)}$

Step 3.4.3.5.2.2.2

Cancel the common factor.

$x = 21 + 25 i + \frac{34 (- y i)}{34 \cdot 1}$

Step 3.4.3.5.2.2.3

Rewrite the expression.

$x = 21 + 25 i + \frac{- y i}{1}$

Step 3.4.3.5.2.2.4

Divide $- y i$ by $1$ .

$x = 21 + 25 i - y i$