Solve for y x+1+2yi=3-6i

$x + 1 + 2 y i = 3 - 6 i$

Step 1

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

Subtract $x$ from both sides of the equation.

$1 + 2 y i = 3 - 6 i - x$

Subtract $1$ from both sides of the equation.

$2 y i = 3 - 6 i - x - 1$

Subtract $1$ from $3$ .

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

Step 2

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $i$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Multiply.

Combine.

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

Multiply $i$ by $1$ .

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

Simplify the denominator.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

Rewrite $- 1 \cdot i$ as $- i$ .

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

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

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

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

Cancel the common factors.

Factor $2$ out of $2 i$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $i$ .

Cancel the common factor.

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

Divide $- 3$ by $1$ .

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

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

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

Multiply.

Combine.

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

Simplify the denominator.

Add parentheses.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Dividing two negative values results in a positive value.

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

Reorder $- i$ and $- 3$ .

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

Solve for y x+1+2yi=3-6i

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