Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Multiply $z$ by $1$ .

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

Step 2

Since $z$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 3

Move all terms containing $z$ to the left side of the equation.

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Step 3.1

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

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

Step 3.2

Subtract $3 i z$ from $2 i z$ .

$z - i z - 3 i = 2 - 4 i$

Step 4

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

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Step 4.1

Add $3 i$ to both sides of the equation.

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

Step 4.2

Add $- 4 i$ and $3 i$ .

$z - i z = 2 - i$

Step 5

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

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Step 5.1

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

$z \cdot 1 - i z = 2 - i$

Step 5.2

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

$z \cdot 1 + z (- i) = 2 - i$

Step 5.3

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

$z (1 - i) = 2 - i$

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $1 - i$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $z$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Combine the numerators over the common denominator.

$z = \frac{2 - i}{1 - i}$

Step 6.3.2

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

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

Step 6.3.3

Multiply.

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Step 6.3.3.1

Combine.

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

Step 6.3.3.2

Simplify the numerator.

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Step 6.3.3.2.1

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

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Step 6.3.3.2.1.1

Apply the distributive property.

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

Step 6.3.3.2.1.2

Apply the distributive property.

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

Step 6.3.3.2.1.3

Apply the distributive property.

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

Step 6.3.3.2.2

Simplify and combine like terms.

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Step 6.3.3.2.2.1

Simplify each term.

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Step 6.3.3.2.2.1.1

Multiply $2$ by $1$ .

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

Step 6.3.3.2.2.1.2

Multiply $- 1$ by $1$ .

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

Step 6.3.3.2.2.1.3

Multiply $- i i$ .

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Step 6.3.3.2.2.1.3.1

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

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

Step 6.3.3.2.2.1.3.2

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

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

Step 6.3.3.2.2.1.3.3

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

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

Step 6.3.3.2.2.1.3.4

Add $1$ and $1$ .

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

Step 6.3.3.2.2.1.4

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

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

Step 6.3.3.2.2.1.5

Multiply $- 1$ by $- 1$ .

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

Step 6.3.3.2.2.2

Add $2$ and $1$ .

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

Step 6.3.3.2.2.3

Subtract $i$ from $2 i$ .

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

Step 6.3.3.3

Simplify the denominator.

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Step 6.3.3.3.1

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

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Step 6.3.3.3.1.1

Apply the distributive property.

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

Step 6.3.3.3.1.2

Apply the distributive property.

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

Step 6.3.3.3.1.3

Apply the distributive property.

$z = \frac{3 + i}{1 \cdot 1 + 1 i - i \cdot 1 - i i}$

Step 6.3.3.3.2

Simplify.

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Step 6.3.3.3.2.1

Multiply $1$ by $1$ .

$z = \frac{3 + i}{1 + 1 i - i \cdot 1 - i i}$

Step 6.3.3.3.2.2

Multiply $- 1$ by $1$ .

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

Step 6.3.3.3.2.3

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

$z = \frac{3 + i}{1 + 1 i - i - (i^{1} i)}$

Step 6.3.3.3.2.4

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

$z = \frac{3 + i}{1 + 1 i - i - (i^{1} i^{1})}$

Step 6.3.3.3.2.5

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

$z = \frac{3 + i}{1 + 1 i - i - i^{1 + 1}}$

Step 6.3.3.3.2.6

Add $1$ and $1$ .

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

Step 6.3.3.3.2.7

Subtract $i$ from $1 i$ .

$z = \frac{3 + i}{1 + 0 - i^{2}}$

Step 6.3.3.3.2.8

Add $1$ and $0$ .

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

Step 6.3.3.3.3

Simplify each term.

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Step 6.3.3.3.3.1

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

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

Step 6.3.3.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 6.3.3.3.4

Add $1$ and $1$ .

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

Step 6.3.4

Split the fraction $\frac{3 + i}{2}$ into two fractions.

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