Algebra Examples

$(x + y i) - (2 - 3 i) = - 6 + 4 i$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$x + y i - 1 \cdot 2 - (- 3 i) = - 6 + 4 i$

Step 1.2

Multiply $- 1$ by $2$ .

$x + y i - 2 - (- 3 i) = - 6 + 4 i$

Step 1.3

Multiply $- 3$ by $- 1$ .

$x + y i - 2 + 3 i = - 6 + 4 i$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the equation.

$y i - 2 + 3 i = - 6 + 4 i - x$

Step 2.2

Add $2$ to both sides of the equation.

$y i + 3 i = - 6 + 4 i - x + 2$

Step 2.3

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

$y i = - 6 + 4 i - x + 2 - 3 i$

Step 2.4

Add $- 6$ and $2$ .

$y i = - 4 + 4 i - x - 3 i$

Step 2.5

Subtract $3 i$ from $4 i$ .

$y i = - 4 - x + i$

Step 3

Divide each term in $y i = - 4 - x + i$ by $i$ and simplify.

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Step 3.1

Divide each term in $y i = - 4 - x + i$ by $i$ .

$\frac{y i}{i} = \frac{- 4}{i} + \frac{- x}{i} + \frac{i}{i}$

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.

$\frac{y i}{i} = \frac{- 4}{i} + \frac{- x}{i} + \frac{i}{i}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4}{i} + \frac{- x}{i} + \frac{i}{i}$

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 $\frac{- 4}{i}$ by the conjugate of $i$ to make the denominator real.

$y = \frac{- 4}{i} \cdot \frac{i}{i} + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Combine.

$y = \frac{- 4 i}{i i} + \frac{- x}{i} + \frac{i}{i}$

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 $1$ .

$y = \frac{- 4 i}{i^{1} i} + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.2.2.2

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

$y = \frac{- 4 i}{i^{1} i^{1}} + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.2.2.3

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

$y = \frac{- 4 i}{i^{1 + 1}} + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.2.2.4

Add $1$ and $1$ .

$y = \frac{- 4 i}{i^{2}} + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.2.2.5

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

$y = \frac{- 4 i}{- 1} + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.3

Move the negative one from the denominator of $\frac{- 4 i}{- 1}$ .

$y = - 1 \cdot (- 4 i) + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.4

Rewrite $- 1 \cdot (- 4 i)$ as $- (- 4 i)$ .

$y = - (- 4 i) + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.5

Multiply $- 4$ by $- 1$ .

$y = 4 i + \frac{- x}{i} + \frac{i}{i}$

Step 3.3.1.6

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

$y = 4 i + \frac{- x}{i} \cdot \frac{i}{i} + \frac{i}{i}$

Step 3.3.1.7

Multiply.

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Step 3.3.1.7.1

Combine.

$y = 4 i + \frac{- x i}{i i} + \frac{i}{i}$

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 $1$ .

$y = 4 i + \frac{- x i}{i^{1} i} + \frac{i}{i}$

Step 3.3.1.7.2.2

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

$y = 4 i + \frac{- x i}{i^{1} i^{1}} + \frac{i}{i}$

Step 3.3.1.7.2.3

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

$y = 4 i + \frac{- x i}{i^{1 + 1}} + \frac{i}{i}$

Step 3.3.1.7.2.4

Add $1$ and $1$ .

$y = 4 i + \frac{- x i}{i^{2}} + \frac{i}{i}$

Step 3.3.1.7.2.5

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

$y = 4 i + \frac{- x i}{- 1} + \frac{i}{i}$

Step 3.3.1.8

Dividing two negative values results in a positive value.

$y = 4 i + \frac{x i}{1} + \frac{i}{i}$

Step 3.3.1.9

Divide $x i$ by $1$ .

$y = 4 i + x i + \frac{i}{i}$

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.

$y = 4 i + x i + \frac{i}{i}$

Step 3.3.1.10.2

Rewrite the expression.

$y = 4 i + x i + 1$

Step 4

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = 4 i + x i + 1$