Precalculus Examples

$(a - 1) + (b + 3) i = 5 + 8 i$

Step 1

Apply the distributive property.

$a - 1 + b i + 3 i = 5 + 8 i$

Step 2

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

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Step 2.1

Subtract $a$ from both sides of the equation.

$- 1 + b i + 3 i = 5 + 8 i - a$

Step 2.2

Add $1$ to both sides of the equation.

$b i + 3 i = 5 + 8 i - a + 1$

Step 2.3

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

$b i = 5 + 8 i - a + 1 - 3 i$

Step 2.4

Add $5$ and $1$ .

$b i = 6 + 8 i - a - 3 i$

Step 2.5

Subtract $3 i$ from $8 i$ .

$b i = 6 - a + 5 i$

Step 3

Divide each term in $b i = 6 - a + 5 i$ by $i$ and simplify.

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Step 3.1

Divide each term in $b i = 6 - a + 5 i$ by $i$ .

$\frac{b i}{i} = \frac{6}{i} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $i$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{b i}{i} = \frac{6}{i} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.2.1.2

Divide $b$ by $1$ .

$b = \frac{6}{i} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply the numerator and denominator of $\frac{6}{i}$ by the conjugate of $i$ to make the denominator real.

$b = \frac{6}{i} \cdot \frac{i}{i} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Combine.

$b = \frac{6 i}{i i} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.2.2

Simplify the denominator.

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Step 3.3.1.2.2.1

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

$b = \frac{6 i}{i^{1} i} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.2.2.2

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

$b = \frac{6 i}{i^{1} i^{1}} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.2.2.3

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

$b = \frac{6 i}{i^{1 + 1}} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.2.2.4

Add $1$ and $1$ .

$b = \frac{6 i}{i^{2}} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.2.2.5

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

$b = \frac{6 i}{- 1} + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.3

Move the negative one from the denominator of $\frac{6 i}{- 1}$ .

$b = - 1 \cdot (6 i) + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.4

Rewrite $- 1 \cdot (6 i)$ as $- (6 i)$ .

$b = - (6 i) + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.5

Multiply $6$ by $- 1$ .

$b = - 6 i + \frac{- a}{i} + \frac{5 i}{i}$

Step 3.3.1.6

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

$b = - 6 i + \frac{- a}{i} \cdot \frac{i}{i} + \frac{5 i}{i}$

Step 3.3.1.7

Multiply.

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Step 3.3.1.7.1

Combine.

$b = - 6 i + \frac{- a i}{i i} + \frac{5 i}{i}$

Step 3.3.1.7.2

Simplify the denominator.

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Step 3.3.1.7.2.1

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

$b = - 6 i + \frac{- a i}{i^{1} i} + \frac{5 i}{i}$

Step 3.3.1.7.2.2

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

$b = - 6 i + \frac{- a i}{i^{1} i^{1}} + \frac{5 i}{i}$

Step 3.3.1.7.2.3

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

$b = - 6 i + \frac{- a i}{i^{1 + 1}} + \frac{5 i}{i}$

Step 3.3.1.7.2.4

Add $1$ and $1$ .

$b = - 6 i + \frac{- a i}{i^{2}} + \frac{5 i}{i}$

Step 3.3.1.7.2.5

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

$b = - 6 i + \frac{- a i}{- 1} + \frac{5 i}{i}$

Step 3.3.1.8

Dividing two negative values results in a positive value.

$b = - 6 i + \frac{a i}{1} + \frac{5 i}{i}$

Step 3.3.1.9

Divide $a i$ by $1$ .

$b = - 6 i + a i + \frac{5 i}{i}$

Step 3.3.1.10

Cancel the common factor of $i$ .

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Step 3.3.1.10.1

Cancel the common factor.

$b = - 6 i + a i + \frac{5 i}{i}$

Step 3.3.1.10.2

Divide $5$ by $1$ .

$b = - 6 i + a i + 5$