Precalculus Examples

$(7 a - 5) - 15 i = 37 - 3 b i$

Step 1

Rewrite the equation as $37 - 3 b i = 7 a - 5 - 15 i$ .

$37 - 3 b i = 7 a - 5 - 15 i$

Step 2

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

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Step 2.1

Subtract $37$ from both sides of the equation.

$- 3 b i = 7 a - 5 - 15 i - 37$

Step 2.2

Subtract $37$ from $- 5$ .

$- 3 b i = 7 a - 42 - 15 i$

Step 3

Divide each term in $- 3 b i = 7 a - 42 - 15 i$ by $- 3 i$ and simplify.

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Step 3.1

Divide each term in $- 3 b i = 7 a - 42 - 15 i$ by $- 3 i$ .

$\frac{- 3 b i}{- 3 i} = \frac{7 a}{- 3 i} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 3 b i}{- 3 i} = \frac{7 a}{- 3 i} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.2.1.2

Rewrite the expression.

$\frac{b i}{i} = \frac{7 a}{- 3 i} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.2.2

Cancel the common factor of $i$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{b i}{i} = \frac{7 a}{- 3 i} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.2.2.2

Divide $b$ by $1$ .

$b = \frac{7 a}{- 3 i} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

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 $\frac{7 a}{- 3 i}$ by the conjugate of $- 3 i$ to make the denominator real.

$b = \frac{7 a}{- 3 i} \cdot \frac{i}{i} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Combine.

$b = \frac{7 a i}{- 3 i i} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.2.2

Simplify the denominator.

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Step 3.3.1.2.2.1

Add parentheses.

$b = \frac{7 a i}{- 3 (i i)} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.2.2.2

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

$b = \frac{7 a i}{- 3 (i^{1} i)} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.2.2.3

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

$b = \frac{7 a i}{- 3 (i^{1} i^{1})} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.2.2.4

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

$b = \frac{7 a i}{- 3 i^{1 + 1}} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.2.2.5

Add $1$ and $1$ .

$b = \frac{7 a i}{- 3 i^{2}} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.2.2.6

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

$b = \frac{7 a i}{- 3 \cdot - 1} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.3

Multiply $- 3$ by $- 1$ .

$b = \frac{7 a i}{3} + \frac{- 42}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.4

Cancel the common factor of $- 42$ and $- 3$ .

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Step 3.3.1.4.1

Factor $- 3$ out of $- 42$ .

$b = \frac{7 a i}{3} + \frac{- 3 (14)}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.4.2

Cancel the common factors.

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Step 3.3.1.4.2.1

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

$b = \frac{7 a i}{3} + \frac{- 3 (14)}{- 3 (i)} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.4.2.2

Cancel the common factor.

$b = \frac{7 a i}{3} + \frac{- 3 \cdot 14}{- 3 i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.4.2.3

Rewrite the expression.

$b = \frac{7 a i}{3} + \frac{14}{i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.5

Multiply the numerator and denominator of $\frac{14}{i}$ by the conjugate of $i$ to make the denominator real.

$b = \frac{7 a i}{3} + \frac{14}{i} \cdot \frac{i}{i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.6

Multiply.

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Step 3.3.1.6.1

Combine.

$b = \frac{7 a i}{3} + \frac{14 i}{i i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.6.2

Simplify the denominator.

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Step 3.3.1.6.2.1

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

$b = \frac{7 a i}{3} + \frac{14 i}{i^{1} i} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.6.2.2

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

$b = \frac{7 a i}{3} + \frac{14 i}{i^{1} i^{1}} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.6.2.3

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

$b = \frac{7 a i}{3} + \frac{14 i}{i^{1 + 1}} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.6.2.4

Add $1$ and $1$ .

$b = \frac{7 a i}{3} + \frac{14 i}{i^{2}} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.6.2.5

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

$b = \frac{7 a i}{3} + \frac{14 i}{- 1} + \frac{- 15 i}{- 3 i}$

Step 3.3.1.7

Move the negative one from the denominator of $\frac{14 i}{- 1}$ .

$b = \frac{7 a i}{3} - 1 \cdot (14 i) + \frac{- 15 i}{- 3 i}$

Step 3.3.1.8

Rewrite $- 1 \cdot (14 i)$ as $- (14 i)$ .

$b = \frac{7 a i}{3} - (14 i) + \frac{- 15 i}{- 3 i}$

Step 3.3.1.9

Multiply $14$ by $- 1$ .

$b = \frac{7 a i}{3} - 14 i + \frac{- 15 i}{- 3 i}$

Step 3.3.1.10

Cancel the common factor of $- 15$ and $- 3$ .

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Step 3.3.1.10.1

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

$b = \frac{7 a i}{3} - 14 i + \frac{- 3 (5 i)}{- 3 i}$

Step 3.3.1.10.2

Cancel the common factors.

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Step 3.3.1.10.2.1

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

$b = \frac{7 a i}{3} - 14 i + \frac{- 3 (5 i)}{- 3 (i)}$

Step 3.3.1.10.2.2

Cancel the common factor.

$b = \frac{7 a i}{3} - 14 i + \frac{- 3 (5 i)}{- 3 i}$

Step 3.3.1.10.2.3

Rewrite the expression.

$b = \frac{7 a i}{3} - 14 i + \frac{5 i}{i}$

Step 3.3.1.11

Cancel the common factor of $i$ .

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Step 3.3.1.11.1

Cancel the common factor.

$b = \frac{7 a i}{3} - 14 i + \frac{5 i}{i}$

Step 3.3.1.11.2

Divide $5$ by $1$ .

$b = \frac{7 a i}{3} - 14 i + 5$