Select the correct answer.

What are the values of [tex] x [/tex] and [tex] y [/tex] that satisfy this equation?

[tex] (x + y i) + (4 + 9 i) = 9 - 4 i [/tex]

A. [tex] x = 5 [/tex] and [tex] y = 13 [/tex]
B. [tex] x = 5 [/tex] and [tex] y = -13 [/tex]
C. [tex] x = 9 [/tex] and [tex] y = -4 [/tex]
D. [tex] x = -9 [/tex] and [tex] y = 4 [/tex]



Answer :

To solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation [tex]\((x + yi) + (4 + 9i) = 9 - 4i\)[/tex], we need to equate the real and imaginary parts on both sides of the equation separately.

The equation given is:
[tex]\[ (x + yi) + (4 + 9i) = 9 - 4i \][/tex]

First, let's separate the real and imaginary parts.

Real part:

The real part on the left side is [tex]\( x + 4 \)[/tex].

The real part on the right side is [tex]\( 9 \)[/tex].

So, we can set up the equation for the real part:
[tex]\[ x + 4 = 9 \][/tex]

Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 9 - 4 \][/tex]
[tex]\[ x = 5 \][/tex]

Imaginary part:

The imaginary part on the left side is [tex]\( yi + 9i \)[/tex].

The imaginary part on the right side is [tex]\( -4i \)[/tex].

So, we can set up the equation for the imaginary part:
[tex]\[ yi + 9i = -4i \][/tex]

Solving for [tex]\( y \)[/tex]:
[tex]\[ y + 9 = -4 \][/tex]
[tex]\[ y = -4 - 9 \][/tex]
[tex]\[ y = -13 \][/tex]

Hence, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the given equation are:
[tex]\[ x = 5 \quad \text{and} \quad y = -13 \][/tex]

Thus, the correct answer is:
[tex]\[ \text{B. } x=5 \text{ and } y=-13 \][/tex]