To determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that the equation
[tex]\[ 22(x + yi) + (28 + 4i) = 72 - 62i \][/tex]
holds true, we need to separate the given equation into its real and imaginary parts and solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] step-by-step.
1. Separate the equation into real and imaginary parts:
We start with:
[tex]\[ 22(x + yi) + (28 + 4i) = 72 - 62i \][/tex]
Distributing the 22 on the left side:
[tex]\[ 22x + 22yi + 28 + 4i = 72 - 62i \][/tex]
2. Combine like terms:
Combine the real parts and the imaginary parts separately on the left side:
[tex]\[ (22x + 28) + (22y + 4)i = 72 - 62i \][/tex]
3. Equate the real and imaginary components:
By comparing the real and imaginary parts from both sides of the equation, we get the two equations:
[tex]\[ 22x + 28 = 72 \tag{Real part} \][/tex]
[tex]\[ 22y + 4 = -62 \tag{Imaginary part} \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 22x + 28 = 72 \][/tex]
[tex]\[ 22x = 72 - 28 \][/tex]
[tex]\[ 22x = 44 \][/tex]
[tex]\[ x = \frac{44}{22} \][/tex]
[tex]\[ x = 2 \][/tex]
5. Solve for [tex]\( y \)[/tex]:
[tex]\[ 22y + 4 = -62 \][/tex]
[tex]\[ 22y = -62 - 4 \][/tex]
[tex]\[ 22y = -66 \][/tex]
[tex]\[ y = \frac{-66}{22} \][/tex]
[tex]\[ y = -3 \][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the given equation are:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = -3 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{B. \, x = 2 \text{ and } y = -3} \][/tex]
