Algebra Examples

$5 - i + (11 - i) z = 40 + 18 i$

Step 1

Apply the distributive property.

$5 - i + 11 z - i z = 40 + 18 i$

Step 2

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

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Step 2.1

Subtract $5$ from both sides of the equation.

$- i + 11 z - i z = 40 + 18 i - 5$

Step 2.2

Add $i$ to both sides of the equation.

$11 z - i z = 40 + 18 i - 5 + i$

Step 2.3

Subtract $5$ from $40$ .

$11 z - i z = 35 + 18 i + i$

Step 2.4

Add $18 i$ and $i$ .

$11 z - i z = 35 + 19 i$

Step 3

Factor $z$ out of $11 z - i z$ .

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Step 3.1

Factor $z$ out of $11 z$ .

$z \cdot 11 - i z = 35 + 19 i$

Step 3.2

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

$z \cdot 11 + z (- i) = 35 + 19 i$

Step 3.3

Factor $z$ out of $z \cdot 11 + z (- i)$ .

$z (11 - i) = 35 + 19 i$

Step 4

Divide each term in $z (11 - i) = 35 + 19 i$ by $11 - i$ and simplify.

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Step 4.1

Divide each term in $z (11 - i) = 35 + 19 i$ by $11 - i$ .

$\frac{z (11 - i)}{11 - i} = \frac{35}{11 - i} + \frac{19 i}{11 - i}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $11 - i$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{z (11 - i)}{11 - i} = \frac{35}{11 - i} + \frac{19 i}{11 - i}$

Step 4.2.1.2

Divide $z$ by $1$ .

$z = \frac{35}{11 - i} + \frac{19 i}{11 - i}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

$z = \frac{35 + 19 i}{11 - i}$

Step 4.3.2

Multiply the numerator and denominator of $\frac{35 + 19 i}{11 - i}$ by the conjugate of $11 - i$ to make the denominator real.

$z = \frac{35 + 19 i}{11 - i} \cdot \frac{11 + i}{11 + i}$

Step 4.3.3

Multiply.

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Step 4.3.3.1

Combine.

$z = \frac{(35 + 19 i) (11 + i)}{(11 - i) (11 + i)}$

Step 4.3.3.2

Simplify the numerator.

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Step 4.3.3.2.1

Expand $(35 + 19 i) (11 + i)$ using the FOIL Method.

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Step 4.3.3.2.1.1

Apply the distributive property.

$z = \frac{35 (11 + i) + 19 i (11 + i)}{(11 - i) (11 + i)}$

Step 4.3.3.2.1.2

Apply the distributive property.

$z = \frac{35 \cdot 11 + 35 i + 19 i (11 + i)}{(11 - i) (11 + i)}$

Step 4.3.3.2.1.3

Apply the distributive property.

$z = \frac{35 \cdot 11 + 35 i + 19 i \cdot 11 + 19 i i}{(11 - i) (11 + i)}$

Step 4.3.3.2.2

Simplify and combine like terms.

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Step 4.3.3.2.2.1

Simplify each term.

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Step 4.3.3.2.2.1.1

Multiply $35$ by $11$ .

$z = \frac{385 + 35 i + 19 i \cdot 11 + 19 i i}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.1.2

Multiply $11$ by $19$ .

$z = \frac{385 + 35 i + 209 i + 19 i i}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.1.3

Multiply $19 i i$ .

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Step 4.3.3.2.2.1.3.1

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

$z = \frac{385 + 35 i + 209 i + 19 (i^{1} i)}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.1.3.2

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

$z = \frac{385 + 35 i + 209 i + 19 (i^{1} i^{1})}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.1.3.3

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

$z = \frac{385 + 35 i + 209 i + 19 i^{1 + 1}}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.1.3.4

Add $1$ and $1$ .

$z = \frac{385 + 35 i + 209 i + 19 i^{2}}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.1.4

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

$z = \frac{385 + 35 i + 209 i + 19 \cdot - 1}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.1.5

Multiply $19$ by $- 1$ .

$z = \frac{385 + 35 i + 209 i - 19}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.2

Subtract $19$ from $385$ .

$z = \frac{366 + 35 i + 209 i}{(11 - i) (11 + i)}$

Step 4.3.3.2.2.3

Add $35 i$ and $209 i$ .

$z = \frac{366 + 244 i}{(11 - i) (11 + i)}$

Step 4.3.3.3

Simplify the denominator.

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Step 4.3.3.3.1

Expand $(11 - i) (11 + i)$ using the FOIL Method.

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Step 4.3.3.3.1.1

Apply the distributive property.

$z = \frac{366 + 244 i}{11 (11 + i) - i (11 + i)}$

Step 4.3.3.3.1.2

Apply the distributive property.

$z = \frac{366 + 244 i}{11 \cdot 11 + 11 i - i (11 + i)}$

Step 4.3.3.3.1.3

Apply the distributive property.

$z = \frac{366 + 244 i}{11 \cdot 11 + 11 i - i \cdot 11 - i i}$

Step 4.3.3.3.2

Simplify.

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Step 4.3.3.3.2.1

Multiply $11$ by $11$ .

$z = \frac{366 + 244 i}{121 + 11 i - i \cdot 11 - i i}$

Step 4.3.3.3.2.2

Multiply $11$ by $- 1$ .

$z = \frac{366 + 244 i}{121 + 11 i - 11 i - i i}$

Step 4.3.3.3.2.3

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

$z = \frac{366 + 244 i}{121 + 11 i - 11 i - (i^{1} i)}$

Step 4.3.3.3.2.4

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

$z = \frac{366 + 244 i}{121 + 11 i - 11 i - (i^{1} i^{1})}$

Step 4.3.3.3.2.5

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

$z = \frac{366 + 244 i}{121 + 11 i - 11 i - i^{1 + 1}}$

Step 4.3.3.3.2.6

Add $1$ and $1$ .

$z = \frac{366 + 244 i}{121 + 11 i - 11 i - i^{2}}$

Step 4.3.3.3.2.7

Subtract $11 i$ from $11 i$ .

$z = \frac{366 + 244 i}{121 + 0 - i^{2}}$

Step 4.3.3.3.2.8

Add $121$ and $0$ .

$z = \frac{366 + 244 i}{121 - i^{2}}$

Step 4.3.3.3.3

Simplify each term.

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Step 4.3.3.3.3.1

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

$z = \frac{366 + 244 i}{121 - - 1}$

Step 4.3.3.3.3.2

Multiply $- 1$ by $- 1$ .

$z = \frac{366 + 244 i}{121 + 1}$

Step 4.3.3.3.4

Add $121$ and $1$ .

$z = \frac{366 + 244 i}{122}$

Step 4.3.4

Cancel the common factor of $366 + 244 i$ and $122$ .

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Step 4.3.4.1

Factor $122$ out of $366$ .

$z = \frac{122 \cdot 3 + 244 i}{122}$

Step 4.3.4.2

Factor $122$ out of $244 i$ .

$z = \frac{122 \cdot 3 + 122 (2 i)}{122}$

Step 4.3.4.3

Factor $122$ out of $122 (3) + 122 (2 i)$ .

$z = \frac{122 (3 + 2 i)}{122}$

Step 4.3.4.4

Cancel the common factors.

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Step 4.3.4.4.1

Factor $122$ out of $122$ .

$z = \frac{122 (3 + 2 i)}{122 (1)}$

Step 4.3.4.4.2

Cancel the common factor.

$z = \frac{122 (3 + 2 i)}{122 \cdot 1}$

Step 4.3.4.4.3

Rewrite the expression.

$z = \frac{3 + 2 i}{1}$

Step 4.3.4.4.4

Divide $3 + 2 i$ by $1$ .

$z = 3 + 2 i$