Solve for y (x+yi)+(4+6i)=7-2i

$(x + y i) + (4 + 6 i) = 7 - 2 i$

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

Subtract $x$ from both sides of the equation.

$y i + 4 + 6 i = 7 - 2 i - x$

Subtract $4$ from both sides of the equation.

$y i + 6 i = 7 - 2 i - x - 4$

Subtract $6 i$ from both sides of the equation.

$y i = 7 - 2 i - x - 4 - 6 i$

Subtract $4$ from $7$ .

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

Subtract $6 i$ from $- 2 i$ .

$y i = 3 - x - 8 i$

Divide each term in $y i = 3 - x - 8 i$ by $i$ and simplify.

Divide each term in $y i = 3 - x - 8 i$ by $i$ .

$\frac{y i}{i} = \frac{3}{i} + \frac{- x}{i} + \frac{- 8 i}{i}$

Simplify the left side.

Cancel the common factor of $i$ .

Cancel the common factor.

$\frac{y i}{i} = \frac{3}{i} + \frac{- x}{i} + \frac{- 8 i}{i}$

Divide $y$ by $1$ .

$y = \frac{3}{i} + \frac{- x}{i} + \frac{- 8 i}{i}$

Simplify the right side.

Simplify each term.

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

$y = \frac{3}{i} \cdot \frac{i}{i} + \frac{- x}{i} + \frac{- 8 i}{i}$

Multiply.

Combine.

$y = \frac{3 i}{i i} + \frac{- x}{i} + \frac{- 8 i}{i}$

Simplify the denominator.

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

$y = \frac{3 i}{i^{1} i} + \frac{- x}{i} + \frac{- 8 i}{i}$

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

$y = \frac{3 i}{i^{1} i^{1}} + \frac{- x}{i} + \frac{- 8 i}{i}$

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

$y = \frac{3 i}{i^{1 + 1}} + \frac{- x}{i} + \frac{- 8 i}{i}$

Add $1$ and $1$ .

$y = \frac{3 i}{i^{2}} + \frac{- x}{i} + \frac{- 8 i}{i}$

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

$y = \frac{3 i}{- 1} + \frac{- x}{i} + \frac{- 8 i}{i}$

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

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

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

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

Multiply $3$ by $- 1$ .

$y = - 3 i + \frac{- x}{i} + \frac{- 8 i}{i}$

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

$y = - 3 i + \frac{- x}{i} \cdot \frac{i}{i} + \frac{- 8 i}{i}$

Multiply.

Combine.

$y = - 3 i + \frac{- x i}{i i} + \frac{- 8 i}{i}$

Simplify the denominator.

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

$y = - 3 i + \frac{- x i}{i^{1} i} + \frac{- 8 i}{i}$

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

$y = - 3 i + \frac{- x i}{i^{1} i^{1}} + \frac{- 8 i}{i}$

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

$y = - 3 i + \frac{- x i}{i^{1 + 1}} + \frac{- 8 i}{i}$

Add $1$ and $1$ .

$y = - 3 i + \frac{- x i}{i^{2}} + \frac{- 8 i}{i}$

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

$y = - 3 i + \frac{- x i}{- 1} + \frac{- 8 i}{i}$

Dividing two negative values results in a positive value.

$y = - 3 i + \frac{x i}{1} + \frac{- 8 i}{i}$

Divide $x i$ by $1$ .

$y = - 3 i + x i + \frac{- 8 i}{i}$

Cancel the common factor of $i$ .

Cancel the common factor.

$y = - 3 i + x i + \frac{- 8 i}{i}$

Divide $- 8$ by $1$ .

$y = - 3 i + x i - 8$

Solve for y (x+yi)+(4+6i)=7-2i

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