To find the real values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for the equation [tex]\(\frac{x + y}{i} + x - y + 4 = 0\)[/tex], let's proceed with the following steps:
1. Rewrite the equation with [tex]\(i\)[/tex]:
The imaginary unit [tex]\(i\)[/tex] is defined as [tex]\(i = \sqrt{-1}\)[/tex]. Knowing this, we also have [tex]\( \frac{1}{i} = -i \)[/tex]. Rewrite the term [tex]\(\frac{x + y}{i}\)[/tex]:
[tex]\[
\frac{x + y}{i} = (x + y) \cdot \left(-i\right) = -i(x + y)
\][/tex]
2. Substitute back into the equation:
Substitute [tex]\(-i(x + y)\)[/tex] back into the equation:
[tex]\[
-i(x + y) + x - y + 4 = 0
\][/tex]
3. Separate the real and imaginary parts:
Next, we need to separate the real and imaginary parts of the equation. Recall that [tex]\(i\)[/tex] is an imaginary unit and [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are real numbers.
[tex]\[
-i(x + y) + x - y + 4 = 0
\][/tex]
To make this separation clear, express it as:
[tex]\[
(x - y + 4) + (-i(x + y)) = 0
\][/tex]
For the equation to hold true, both the real part and the imaginary part must individually equal zero.
4. Set the real part to zero:
The real part of the equation is:
[tex]\[
x - y + 4 = 0
\][/tex]
Simplify this to find a relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
x - y = -4 \quad \text{(Equation 1)}
\][/tex]
5. Set the imaginary part to zero:
The coefficient of the imaginary unit [tex]\(i\)[/tex] must also equal zero:
[tex]\[
-i(x + y) = 0 \implies x + y = 0 \quad \text{(Equation 2)}
\][/tex]
6. Solve the system of equations:
Now, solve the system of linear equations:
[tex]\[
\begin{cases}
x - y = -4 \\
x + y = 0
\end{cases}
\][/tex]
a. From Equation 2, solve for [tex]\(x\)[/tex]:
[tex]\[
x = -y
\][/tex]
b. Substitute [tex]\(x = -y\)[/tex] into Equation 1:
[tex]\[
-y - y = -4
\][/tex]
Simplify:
[tex]\[
-2y = -4 \implies y = 2
\][/tex]
c. Substitute [tex]\(y = 2\)[/tex] back into Equation 2 to find [tex]\(x\)[/tex]:
[tex]\[
x + 2 = 0 \implies x = -2
\][/tex]
7. Conclusion:
Therefore, the real values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy the equation [tex]\(\frac{x+y}{i}+x-y+4=0\)[/tex] are:
[tex]\[
x = -2 \quad \text{and} \quad y = 2
\][/tex]
