Precalculus Examples

$- 5 x + 3 i y = 10 - 9 i$

Step 1

Add $5 x$ to both sides of the equation.

$3 i y = 10 - 9 i + 5 x$

Step 2

Divide each term in $3 i y = 10 - 9 i + 5 x$ by $3 i$ and simplify.

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Step 2.1

Divide each term in $3 i y = 10 - 9 i + 5 x$ by $3 i$ .

$\frac{3 i y}{3 i} = \frac{10}{3 i} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 i y}{3 i} = \frac{10}{3 i} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.2.1.2

Rewrite the expression.

$\frac{i y}{i} = \frac{10}{3 i} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.2.2

Cancel the common factor of $i$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{i y}{i} = \frac{10}{3 i} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.2.2.2

Divide $y$ by $1$ .

$y = \frac{10}{3 i} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply the numerator and denominator of $\frac{10}{3 i}$ by the conjugate of $3 i$ to make the denominator real.

$y = \frac{10}{3 i} \cdot \frac{i}{i} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.2

Multiply.

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Step 2.3.1.2.1

Combine.

$y = \frac{10 i}{3 i i} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.2.2

Simplify the denominator.

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Step 2.3.1.2.2.1

Add parentheses.

$y = \frac{10 i}{3 (i i)} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.2.2.2

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

$y = \frac{10 i}{3 (i^{1} i)} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.2.2.3

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

$y = \frac{10 i}{3 (i^{1} i^{1})} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.2.2.4

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

$y = \frac{10 i}{3 i^{1 + 1}} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.2.2.5

Add $1$ and $1$ .

$y = \frac{10 i}{3 i^{2}} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.2.2.6

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

$y = \frac{10 i}{3 \cdot - 1} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.3

Multiply $3$ by $- 1$ .

$y = \frac{10 i}{- 3} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.4

Move the negative in front of the fraction.

$y = - \frac{10 i}{3} + \frac{- 9 i}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.5

Cancel the common factor of $- 9$ and $3$ .

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Step 2.3.1.5.1

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

$y = - \frac{10 i}{3} + \frac{3 (- 3 i)}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.5.2

Cancel the common factors.

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Step 2.3.1.5.2.1

Factor $3$ out of $3 i$ .

$y = - \frac{10 i}{3} + \frac{3 (- 3 i)}{3 (i)} + \frac{5 x}{3 i}$

Step 2.3.1.5.2.2

Cancel the common factor.

$y = - \frac{10 i}{3} + \frac{3 (- 3 i)}{3 i} + \frac{5 x}{3 i}$

Step 2.3.1.5.2.3

Rewrite the expression.

$y = - \frac{10 i}{3} + \frac{- 3 i}{i} + \frac{5 x}{3 i}$

Step 2.3.1.6

Cancel the common factor of $i$ .

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Step 2.3.1.6.1

Cancel the common factor.

$y = - \frac{10 i}{3} + \frac{- 3 i}{i} + \frac{5 x}{3 i}$

Step 2.3.1.6.2

Divide $- 3$ by $1$ .

$y = - \frac{10 i}{3} - 3 + \frac{5 x}{3 i}$

Step 2.3.1.7

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

$y = - \frac{10 i}{3} - 3 + \frac{5 x}{3 i} \cdot \frac{i}{i}$

Step 2.3.1.8

Multiply.

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Step 2.3.1.8.1

Combine.

$y = - \frac{10 i}{3} - 3 + \frac{5 x i}{3 i i}$

Step 2.3.1.8.2

Simplify the denominator.

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Step 2.3.1.8.2.1

Add parentheses.

$y = - \frac{10 i}{3} - 3 + \frac{5 x i}{3 (i i)}$

Step 2.3.1.8.2.2

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

$y = - \frac{10 i}{3} - 3 + \frac{5 x i}{3 (i^{1} i)}$

Step 2.3.1.8.2.3

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

$y = - \frac{10 i}{3} - 3 + \frac{5 x i}{3 (i^{1} i^{1})}$

Step 2.3.1.8.2.4

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

$y = - \frac{10 i}{3} - 3 + \frac{5 x i}{3 i^{1 + 1}}$

Step 2.3.1.8.2.5

Add $1$ and $1$ .

$y = - \frac{10 i}{3} - 3 + \frac{5 x i}{3 i^{2}}$

Step 2.3.1.8.2.6

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

$y = - \frac{10 i}{3} - 3 + \frac{5 x i}{3 \cdot - 1}$

Step 2.3.1.9

Multiply $3$ by $- 1$ .

$y = - \frac{10 i}{3} - 3 + \frac{5 x i}{- 3}$

Step 2.3.1.10

Move the negative in front of the fraction.

$y = - \frac{10 i}{3} - 3 - \frac{5 x i}{3}$

Step 2.3.2

Reorder $- \frac{10 i}{3}$ and $- 3$ .

$y = - 3 - \frac{10 i}{3} - \frac{5 x i}{3}$