Algebra Examples

$(x + 2) + (2 y - 1) i = - 1 + 5 i$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$x + 2 + 2 y i - 1 i = - 1 + 5 i$

Step 1.2

Rewrite $- 1 i$ as $- i$ .

$x + 2 + 2 y i - i = - 1 + 5 i$

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.

$2 + 2 y i - i = - 1 + 5 i - x$

Step 2.2

Subtract $2$ from both sides of the equation.

$2 y i - i = - 1 + 5 i - x - 2$

Step 2.3

Add $i$ to both sides of the equation.

$2 y i = - 1 + 5 i - x - 2 + i$

Step 2.4

Subtract $2$ from $- 1$ .

$2 y i = - 3 + 5 i - x + i$

Step 2.5

Add $5 i$ and $i$ .

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

Step 3

Divide each term in $2 y i = - 3 - x + 6 i$ by $2 i$ and simplify.

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Step 3.1

Divide each term in $2 y i = - 3 - x + 6 i$ by $2 i$ .

$\frac{2 y i}{2 i} = \frac{- 3}{2 i} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 y i}{2 i} = \frac{- 3}{2 i} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.2.1.2

Rewrite the expression.

$\frac{y i}{i} = \frac{- 3}{2 i} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.2.2

Cancel the common factor of $i$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{y i}{i} = \frac{- 3}{2 i} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 3}{2 i} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

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 $\frac{- 3}{2 i}$ by the conjugate of $2 i$ to make the denominator real.

$y = \frac{- 3}{2 i} \cdot \frac{i}{i} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Combine.

$y = \frac{- 3 i}{2 i i} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.2.2

Simplify the denominator.

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Step 3.3.1.2.2.1

Add parentheses.

$y = \frac{- 3 i}{2 (i i)} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.2.2.2

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

$y = \frac{- 3 i}{2 (i^{1} i)} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.2.2.3

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

$y = \frac{- 3 i}{2 (i^{1} i^{1})} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.2.2.4

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

$y = \frac{- 3 i}{2 i^{1 + 1}} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.2.2.5

Add $1$ and $1$ .

$y = \frac{- 3 i}{2 i^{2}} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.2.2.6

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

$y = \frac{- 3 i}{2 \cdot - 1} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.3

Multiply $2$ by $- 1$ .

$y = \frac{- 3 i}{- 2} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.4

Dividing two negative values results in a positive value.

$y = \frac{3 i}{2} + \frac{- x}{2 i} + \frac{6 i}{2 i}$

Step 3.3.1.5

Multiply the numerator and denominator of $\frac{- x}{2 i}$ by the conjugate of $2 i$ to make the denominator real.

$y = \frac{3 i}{2} + \frac{- x}{2 i} \cdot \frac{i}{i} + \frac{6 i}{2 i}$

Step 3.3.1.6

Multiply.

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Step 3.3.1.6.1

Combine.

$y = \frac{3 i}{2} + \frac{- x i}{2 i i} + \frac{6 i}{2 i}$

Step 3.3.1.6.2

Simplify the denominator.

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Step 3.3.1.6.2.1

Add parentheses.

$y = \frac{3 i}{2} + \frac{- x i}{2 (i i)} + \frac{6 i}{2 i}$

Step 3.3.1.6.2.2

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

$y = \frac{3 i}{2} + \frac{- x i}{2 (i^{1} i)} + \frac{6 i}{2 i}$

Step 3.3.1.6.2.3

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

$y = \frac{3 i}{2} + \frac{- x i}{2 (i^{1} i^{1})} + \frac{6 i}{2 i}$

Step 3.3.1.6.2.4

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

$y = \frac{3 i}{2} + \frac{- x i}{2 i^{1 + 1}} + \frac{6 i}{2 i}$

Step 3.3.1.6.2.5

Add $1$ and $1$ .

$y = \frac{3 i}{2} + \frac{- x i}{2 i^{2}} + \frac{6 i}{2 i}$

Step 3.3.1.6.2.6

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

$y = \frac{3 i}{2} + \frac{- x i}{2 \cdot - 1} + \frac{6 i}{2 i}$

Step 3.3.1.7

Dividing two negative values results in a positive value.

$y = \frac{3 i}{2} + \frac{x i}{2} + \frac{6 i}{2 i}$

Step 3.3.1.8

Cancel the common factor of $6$ and $2$ .

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Step 3.3.1.8.1

Factor $2$ out of $6 i$ .

$y = \frac{3 i}{2} + \frac{x i}{2} + \frac{2 (3 i)}{2 i}$

Step 3.3.1.8.2

Cancel the common factors.

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Step 3.3.1.8.2.1

Factor $2$ out of $2 i$ .

$y = \frac{3 i}{2} + \frac{x i}{2} + \frac{2 (3 i)}{2 (i)}$

Step 3.3.1.8.2.2

Cancel the common factor.

$y = \frac{3 i}{2} + \frac{x i}{2} + \frac{2 (3 i)}{2 i}$

Step 3.3.1.8.2.3

Rewrite the expression.

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

Step 3.3.1.9

Cancel the common factor of $i$ .

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Step 3.3.1.9.1

Cancel the common factor.

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

Step 3.3.1.9.2

Divide $3$ by $1$ .

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