Solve for x (4-6i)/x=8-2i

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

Step 1

Find the LCD of the terms in the equation.

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

$x, 1, 1$

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.

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

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Solve the equation.

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the left side.

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

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

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4 - i$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

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

Factor $2$ out of $4$ .

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

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

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

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

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $4$ .

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

Multiply $4$ by $- 3$ .

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

Multiply $- 3 i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 3$ by $- 1$ .

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

Add $8$ and $3$ .

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

Subtract $12 i$ from $2 i$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $4$ by $4$ .

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

Multiply $4$ by $- 1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $4 i$ from $4 i$ .

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

Add $16$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $16$ and $1$ .

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

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

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

Move the negative in front of the fraction.

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

Solve for x (4-6i)/x=8-2i

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