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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]
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].
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