Question

question 2
type the correct answer in each box. use numerals instead of words.
find the values of x and y in these equations.
$(x + yi) + (4 + 6i) = 7 - 2i$ (equation a)
$(x + yi) - (-8 + 11i) = 5 + 9i$ (equation b)
in equation a, $x = \square$ and $y = \square$.
in equation b, $x = \square$ and $y = \square$.

Explanation:

Equation A

Step1: Combine like terms (real and imaginary)

For the equation \((x + yi) + (4 + 6i) = 7 - 2i\), we can rewrite it by combining the real parts and the imaginary parts separately. The real parts are \(x\) and \(4\), and the imaginary parts are \(y\) (from \(yi\)) and \(6\) (from \(6i\)). So we get:
\((x + 4) + (y + 6)i = 7 - 2i\)

Step2: Equate real and imaginary parts

For two complex numbers to be equal, their real parts must be equal and their imaginary parts must be equal.

• For the real parts: \(x + 4 = 7\)

Solving for \(x\), we subtract \(4\) from both sides: \(x = 7 - 4 = 3\)

• For the imaginary parts: \(y + 6 = -2\)

Solving for \(y\), we subtract \(6\) from both sides: \(y = -2 - 6 = -8\)

Equation B

Step1: Simplify the left - hand side

First, simplify \((x + yi)-(-8 + 11i)\). We distribute the negative sign: \(x + yi + 8 - 11i=(x + 8)+(y - 11)i\)
So the equation \((x + yi)-(-8 + 11i)=5 + 9i\) becomes \((x + 8)+(y - 11)i = 5 + 9i\)

Step2: Equate real and imaginary parts

• For the real parts: \(x + 8 = 5\)

Solving for \(x\), we subtract \(8\) from both sides: \(x = 5 - 8=-3\)

• For the imaginary parts: \(y - 11 = 9\)

Solving for \(y\), we add \(11\) to both sides: \(y=9 + 11 = 20\)

Answer:

In equation A, \(x = \boldsymbol{3}\) and \(y=\boldsymbol{-8}\).
In equation B, \(x=\boldsymbol{-3}\) and \(y = \boldsymbol{20}\).