Algebra Examples

$\frac{4 - 6 i}{x} = 8 - 2 i$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1, 1$

Step 1.2

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $\frac{4 - 6 i}{x} = 8 - 2 i$ by $x$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{4 - 6 i}{x} = 8 - 2 i$ by $x$ .

$\frac{4 - 6 i}{x} x = 8 x - 2 i x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 - 6 i}{x} x = 8 x - 2 i x$

Step 2.2.1.2

Rewrite the expression.

$4 - 6 i = 8 x - 2 i x$

Step 3

Solve the equation.

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Step 3.1

Rewrite the equation as $8 x - 2 i x = 4 - 6 i$ .

$8 x - 2 i x = 4 - 6 i$

Step 3.2

Factor $2 x$ out of $8 x - 2 i x$ .

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Step 3.2.1

Factor $2 x$ out of $8 x$ .

$2 x (4) - 2 i x = 4 - 6 i$

Step 3.2.2

Factor $2 x$ out of $- 2 i x$ .

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

Step 3.2.3

Factor $2 x$ out of $2 x (4) + 2 x (- i)$ .

$2 x (4 - i) = 4 - 6 i$

Step 3.3

Divide each term in $2 x (4 - i) = 4 - 6 i$ by $8 - 2 i$ and simplify.

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Step 3.3.1

Divide each term in $2 x (4 - i) = 4 - 6 i$ by $8 - 2 i$ .

$\frac{2 x (4 - i)}{8 - 2 i} = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ and $8 - 2 i$ .

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Step 3.3.2.1.1

Factor $2$ out of $2 x (4 - i)$ .

$\frac{2 (x (4 - i))}{8 - 2 i} = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.2.1.2

Cancel the common factors.

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Step 3.3.2.1.2.1

Factor $2$ out of $8$ .

$\frac{2 (x (4 - i))}{2 (4) - 2 i} = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.2.1.2.2

Factor $2$ out of $- 2 i$ .

$\frac{2 (x (4 - i))}{2 (4) + 2 (- i)} = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.2.1.2.3

Factor $2$ out of $2 (4) + 2 (- i)$ .

$\frac{2 (x (4 - i))}{2 (4 - i)} = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.2.1.2.4

Cancel the common factor.

$\frac{2 (x (4 - i))}{2 (4 - i)} = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.2.1.2.5

Rewrite the expression.

$\frac{x (4 - i)}{4 - i} = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.2.2

Cancel the common factor of $4 - i$ .

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Step 3.3.2.2.1

Cancel the common factor.

$\frac{x (4 - i)}{4 - i} = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{4}{8 - 2 i} + \frac{- 6 i}{8 - 2 i}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Combine the numerators over the common denominator.

$x = \frac{4 - 6 i}{8 - 2 i}$

Step 3.3.3.2

Cancel the common factor of $4 - 6 i$ and $8 - 2 i$ .

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Step 3.3.3.2.1

Factor $2$ out of $4$ .

$x = \frac{2 (2) - 6 i}{8 - 2 i}$

Step 3.3.3.2.2

Factor $2$ out of $- 6 i$ .

$x = \frac{2 (2) + 2 (- 3 i)}{8 - 2 i}$

Step 3.3.3.2.3

Factor $2$ out of $2 (2) + 2 (- 3 i)$ .

$x = \frac{2 (2 - 3 i)}{8 - 2 i}$

Step 3.3.3.2.4

Cancel the common factors.

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Step 3.3.3.2.4.1

Factor $2$ out of $8$ .

$x = \frac{2 (2 - 3 i)}{2 \cdot 4 - 2 i}$

Step 3.3.3.2.4.2

Factor $2$ out of $- 2 i$ .

$x = \frac{2 (2 - 3 i)}{2 \cdot 4 + 2 (- i)}$

Step 3.3.3.2.4.3

Factor $2$ out of $2 (4) + 2 (- i)$ .

$x = \frac{2 (2 - 3 i)}{2 (4 - i)}$

Step 3.3.3.2.4.4

Cancel the common factor.

$x = \frac{2 (2 - 3 i)}{2 (4 - i)}$

Step 3.3.3.2.4.5

Rewrite the expression.

$x = \frac{2 - 3 i}{4 - i}$

Step 3.3.3.3

Multiply the numerator and denominator of $\frac{2 - 3 i}{4 - i}$ by the conjugate of $4 - i$ to make the denominator real.

$x = \frac{2 - 3 i}{4 - i} \cdot \frac{4 + i}{4 + i}$

Step 3.3.3.4

Multiply.

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Step 3.3.3.4.1

Combine.

$x = \frac{(2 - 3 i) (4 + i)}{(4 - i) (4 + i)}$

Step 3.3.3.4.2

Simplify the numerator.

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Step 3.3.3.4.2.1

Expand $(2 - 3 i) (4 + i)$ using the FOIL Method.

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Step 3.3.3.4.2.1.1

Apply the distributive property.

$x = \frac{2 (4 + i) - 3 i (4 + i)}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.1.2

Apply the distributive property.

$x = \frac{2 \cdot 4 + 2 i - 3 i (4 + i)}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.1.3

Apply the distributive property.

$x = \frac{2 \cdot 4 + 2 i - 3 i \cdot 4 - 3 i i}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2

Simplify and combine like terms.

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Step 3.3.3.4.2.2.1

Simplify each term.

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Step 3.3.3.4.2.2.1.1

Multiply $2$ by $4$ .

$x = \frac{8 + 2 i - 3 i \cdot 4 - 3 i i}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.1.2

Multiply $4$ by $- 3$ .

$x = \frac{8 + 2 i - 12 i - 3 i i}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.1.3

Multiply $- 3 i i$ .

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Step 3.3.3.4.2.2.1.3.1

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

$x = \frac{8 + 2 i - 12 i - 3 (i^{1} i)}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.1.3.2

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

$x = \frac{8 + 2 i - 12 i - 3 (i^{1} i^{1})}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.1.3.3

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

$x = \frac{8 + 2 i - 12 i - 3 i^{1 + 1}}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.1.3.4

Add $1$ and $1$ .

$x = \frac{8 + 2 i - 12 i - 3 i^{2}}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.1.4

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

$x = \frac{8 + 2 i - 12 i - 3 \cdot - 1}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.1.5

Multiply $- 3$ by $- 1$ .

$x = \frac{8 + 2 i - 12 i + 3}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.2

Add $8$ and $3$ .

$x = \frac{11 + 2 i - 12 i}{(4 - i) (4 + i)}$

Step 3.3.3.4.2.2.3

Subtract $12 i$ from $2 i$ .

$x = \frac{11 - 10 i}{(4 - i) (4 + i)}$

Step 3.3.3.4.3

Simplify the denominator.

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Step 3.3.3.4.3.1

Expand $(4 - i) (4 + i)$ using the FOIL Method.

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Step 3.3.3.4.3.1.1

Apply the distributive property.

$x = \frac{11 - 10 i}{4 (4 + i) - i (4 + i)}$

Step 3.3.3.4.3.1.2

Apply the distributive property.

$x = \frac{11 - 10 i}{4 \cdot 4 + 4 i - i (4 + i)}$

Step 3.3.3.4.3.1.3

Apply the distributive property.

$x = \frac{11 - 10 i}{4 \cdot 4 + 4 i - i \cdot 4 - i i}$

Step 3.3.3.4.3.2

Simplify.

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Step 3.3.3.4.3.2.1

Multiply $4$ by $4$ .

$x = \frac{11 - 10 i}{16 + 4 i - i \cdot 4 - i i}$

Step 3.3.3.4.3.2.2

Multiply $4$ by $- 1$ .

$x = \frac{11 - 10 i}{16 + 4 i - 4 i - i i}$

Step 3.3.3.4.3.2.3

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

$x = \frac{11 - 10 i}{16 + 4 i - 4 i - (i^{1} i)}$

Step 3.3.3.4.3.2.4

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

$x = \frac{11 - 10 i}{16 + 4 i - 4 i - (i^{1} i^{1})}$

Step 3.3.3.4.3.2.5

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

$x = \frac{11 - 10 i}{16 + 4 i - 4 i - i^{1 + 1}}$

Step 3.3.3.4.3.2.6

Add $1$ and $1$ .

$x = \frac{11 - 10 i}{16 + 4 i - 4 i - i^{2}}$

Step 3.3.3.4.3.2.7

Subtract $4 i$ from $4 i$ .

$x = \frac{11 - 10 i}{16 + 0 - i^{2}}$

Step 3.3.3.4.3.2.8

Add $16$ and $0$ .

$x = \frac{11 - 10 i}{16 - i^{2}}$

Step 3.3.3.4.3.3

Simplify each term.

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Step 3.3.3.4.3.3.1

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

$x = \frac{11 - 10 i}{16 - - 1}$

Step 3.3.3.4.3.3.2

Multiply $- 1$ by $- 1$ .

$x = \frac{11 - 10 i}{16 + 1}$

Step 3.3.3.4.3.4

Add $16$ and $1$ .

$x = \frac{11 - 10 i}{17}$

Step 3.3.3.5

Split the fraction $\frac{11 - 10 i}{17}$ into two fractions.

$x = \frac{11}{17} + \frac{- 10 i}{17}$

Step 3.3.3.6

Move the negative in front of the fraction.

$x = \frac{11}{17} - \frac{10 i}{17}$