Select the correct answer.

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

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

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



Answer :

To determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the equation [tex]\((x + yi) + (4 + 9i) = 9 - 4i\)[/tex], we can separate the equation into real and imaginary parts and solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

1. Start with the given equation:
[tex]\[ (x + yi) + (4 + 9i) = 9 - 4i \][/tex]

2. Combine the real parts and the imaginary parts separately:
[tex]\[ (x + 4) + (y + 9)i = 9 - 4i \][/tex]

3. Set the real parts equal to each other and the imaginary parts equal to each other:
[tex]\[ x + 4 = 9 \quad \text{and} \quad y + 9 = -4 \][/tex]

4. Solve for [tex]\( x \)[/tex] from the real parts equation:
[tex]\[ x + 4 = 9 \][/tex]
[tex]\[ x = 9 - 4 \][/tex]
[tex]\[ x = 5 \][/tex]

5. Solve for [tex]\( y \)[/tex] from the imaginary parts equation:
[tex]\[ y + 9 = -4 \][/tex]
[tex]\[ y = -4 - 9 \][/tex]
[tex]\[ y = -13 \][/tex]

Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the equation are [tex]\( x = 5 \)[/tex] and [tex]\( y = -13 \)[/tex].

The correct answer is:

C. [tex]\(x=5\)[/tex] and [tex]\(y=-13\)[/tex]