Algebra Examples

$x + 3 i = 9 - y i$

Step 1

Rewrite the equation as $9 - y i = x + 3 i$ .

$9 - y i = x + 3 i$

Step 2

Subtract $9$ from both sides of the equation.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2

Cancel the common factor of $i$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Divide $y$ by $1$ .

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

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

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

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Combine.

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

Step 3.3.1.2.2

Simplify the denominator.

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Step 3.3.1.2.2.1

Add parentheses.

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

Step 3.3.1.2.2.2

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

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

Step 3.3.1.2.2.3

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

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

Step 3.3.1.2.2.4

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

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

Step 3.3.1.2.2.5

Add $1$ and $1$ .

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

Step 3.3.1.2.2.6

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

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

Step 3.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.4

Divide $x i$ by $1$ .

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

Step 3.3.1.5

Cancel the common factor of $i$ .

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Step 3.3.1.5.1

Cancel the common factor.

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

Step 3.3.1.5.2

Rewrite the expression.

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

Step 3.3.1.5.3

Move the negative one from the denominator of $\frac{3}{- 1}$ .

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

Step 3.3.1.6

Multiply $- 1$ by $3$ .

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

Step 3.3.1.7

Multiply the numerator and denominator of $\frac{- 9}{- i}$ by the conjugate of $- i$ to make the denominator real.

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

Step 3.3.1.8

Multiply.

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Step 3.3.1.8.1

Combine.

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

Step 3.3.1.8.2

Simplify the denominator.

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Step 3.3.1.8.2.1

Add parentheses.

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

Step 3.3.1.8.2.2

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

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

Step 3.3.1.8.2.3

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

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

Step 3.3.1.8.2.4

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

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

Step 3.3.1.8.2.5

Add $1$ and $1$ .

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

Step 3.3.1.8.2.6

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

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

Step 3.3.1.9

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.10

Divide $- 9 i$ by $1$ .

$y = x i - 3 - 9 i$