Precalculus Examples

$- 2 x - 1 + 2 y i = 4 x - 5 i$

Step 1

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

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Step 1.1

Add $2 x$ to both sides of the equation.

$- 1 + 2 y i = 4 x - 5 i + 2 x$

Step 1.2

Add $1$ to both sides of the equation.

$2 y i = 4 x - 5 i + 2 x + 1$

Step 1.3

Add $4 x$ and $2 x$ .

$2 y i = 6 x - 5 i + 1$

Step 2

Divide each term in $2 y i = 6 x - 5 i + 1$ by $2 i$ and simplify.

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Step 2.1

Divide each term in $2 y i = 6 x - 5 i + 1$ by $2 i$ .

$\frac{2 y i}{2 i} = \frac{6 x}{2 i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 y i}{2 i} = \frac{6 x}{2 i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.2.1.2

Rewrite the expression.

$\frac{y i}{i} = \frac{6 x}{2 i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.2.2

Cancel the common factor of $i$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{y i}{i} = \frac{6 x}{2 i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.2.2.2

Divide $y$ by $1$ .

$y = \frac{6 x}{2 i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

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

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

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Step 2.3.1.1.1

Factor $2$ out of $6 x$ .

$y = \frac{2 (3 x)}{2 i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.1.2

Cancel the common factors.

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Step 2.3.1.1.2.1

Factor $2$ out of $2 i$ .

$y = \frac{2 (3 x)}{2 (i)} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 (3 x)}{2 i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.1.2.3

Rewrite the expression.

$y = \frac{3 x}{i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.2

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

$y = \frac{3 x}{i} \cdot \frac{i}{i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.3

Multiply.

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Step 2.3.1.3.1

Combine.

$y = \frac{3 x i}{i i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.3.2

Simplify the denominator.

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Step 2.3.1.3.2.1

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

$y = \frac{3 x i}{i^{1} i} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.3.2.2

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

$y = \frac{3 x i}{i^{1} i^{1}} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.3.2.3

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

$y = \frac{3 x i}{i^{1 + 1}} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.3.2.4

Add $1$ and $1$ .

$y = \frac{3 x i}{i^{2}} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.3.2.5

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

$y = \frac{3 x i}{- 1} + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.4

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

$y = - 1 \cdot (3 x i) + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.5

Rewrite $- 1 \cdot (3 x i)$ as $- (3 x i)$ .

$y = - (3 x i) + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.6

Multiply $3$ by $- 1$ .

$y = - 3 (x i) + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.7

Cancel the common factor of $i$ .

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Step 2.3.1.7.1

Cancel the common factor.

$y = - 3 x i + \frac{- 5 i}{2 i} + \frac{1}{2 i}$

Step 2.3.1.7.2

Rewrite the expression.

$y = - 3 x i + \frac{- 5}{2} + \frac{1}{2 i}$

Step 2.3.1.8

Move the negative in front of the fraction.

$y = - 3 x i - \frac{5}{2} + \frac{1}{2 i}$

Step 2.3.1.9

Multiply the numerator and denominator of $\frac{1}{2 i}$ by the conjugate of $2 i$ to make the denominator real.

$y = - 3 x i - \frac{5}{2} + \frac{1}{2 i} \cdot \frac{i}{i}$

Step 2.3.1.10

Multiply.

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Step 2.3.1.10.1

Combine.

$y = - 3 x i - \frac{5}{2} + \frac{1 i}{2 i i}$

Step 2.3.1.10.2

Multiply $i$ by $1$ .

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

Step 2.3.1.10.3

Simplify the denominator.

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Step 2.3.1.10.3.1

Add parentheses.

$y = - 3 x i - \frac{5}{2} + \frac{i}{2 (i i)}$

Step 2.3.1.10.3.2

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

$y = - 3 x i - \frac{5}{2} + \frac{i}{2 (i^{1} i)}$

Step 2.3.1.10.3.3

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

$y = - 3 x i - \frac{5}{2} + \frac{i}{2 (i^{1} i^{1})}$

Step 2.3.1.10.3.4

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

$y = - 3 x i - \frac{5}{2} + \frac{i}{2 i^{1 + 1}}$

Step 2.3.1.10.3.5

Add $1$ and $1$ .

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

Step 2.3.1.10.3.6

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

$y = - 3 x i - \frac{5}{2} + \frac{i}{2 \cdot - 1}$

Step 2.3.1.11

Multiply $2$ by $- 1$ .

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

Step 2.3.1.12

Move the negative in front of the fraction.

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

Step 2.3.2

To write $- 3 x i$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - 3 x i \cdot \frac{2}{2} - \frac{i}{2} - \frac{5}{2}$

Step 2.3.3

Combine $- 3 x i$ and $\frac{2}{2}$ .

$y = \frac{- 3 x i \cdot 2}{2} - \frac{i}{2} - \frac{5}{2}$

Step 2.3.4

Combine the numerators over the common denominator.

$y = \frac{- 3 x i \cdot 2 - i}{2} - \frac{5}{2}$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Multiply $2$ by $- 3$ .

$y = \frac{- 6 x i - i}{2} - \frac{5}{2}$

Step 2.3.5.2

Reorder terms.

$y = \frac{- 6 i x - i}{2} - \frac{5}{2}$

Step 2.3.6

Combine the numerators over the common denominator.

$y = \frac{- 6 i x - i - 5}{2}$

Step 2.3.7

Reorder terms.

$y = \frac{- 6 i x - 5 - i}{2}$

Step 2.3.8

Factor $- 1$ out of $- 6 i x$ .

$y = \frac{- (6 i x) - 5 - i}{2}$

Step 2.3.9

Rewrite $- 5$ as $- 1 (5)$ .

$y = \frac{- (6 i x) - 1 (5) - i}{2}$

Step 2.3.10

Factor $- 1$ out of $- (6 i x) - 1 (5)$ .

$y = \frac{- (6 i x + 5) - i}{2}$

Step 2.3.11

Factor $- 1$ out of $- i$ .

$y = \frac{- (6 i x + 5) - (i)}{2}$

Step 2.3.12

Factor $- 1$ out of $- (6 i x + 5) - (i)$ .

$y = \frac{- (6 i x + 5 + i)}{2}$

Step 2.3.13

Simplify the expression.

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Step 2.3.13.1

Rewrite $- (6 i x + 5 + i)$ as $- 1 (6 i x + 5 + i)$ .

$y = \frac{- 1 (6 i x + 5 + i)}{2}$

Step 2.3.13.2

Move the negative in front of the fraction.

$y = - \frac{6 i x + 5 + i}{2}$