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The system of equations you've provided is:

\[
5X \left( 1 + \frac{1}{X^2 + Y^2} \right) = 12
\]
\[
5Y \left( 1 - \frac{1}{X^2 + Y^2} \right) = 4
\]

Let's go through the solution step by step.

### 1. First equation:
\[
5X \left( 1 + \frac{1}{X^2 + Y^2} \right) = 12
\]
Divide both sides by 5:
\[
X \left( 1 + \frac{1}{X^2 + Y^2} \right) = \frac{12}{5}
\]
Simplify:
\[
X + \frac{X}{X^2 + Y^2} = \frac{12}{5}
\]

### 2. Second equation:
\[
5Y \left( 1 - \frac{1}{X^2 + Y^2} \right) = 4
\]
Divide both sides by 5:
\[
Y \left( 1 - \frac{1}{X^2 + Y^2} \right) = \frac{4}{5}
\]
Simplify:
\[
Y - \frac{Y}{X^2 + Y^2} = \frac{4}{5}
\]

Now we have two simplified equations. Let’s solve for \( X \) and \( Y \). I'll continue the process to find the solution.The system of equations has the following solutions:

1. \( X = 0.4 \), \( Y = -0.2 \)
2. \( X = 2 \), \( Y = 1 \)
3. \( X = 1.2 - 0.6i \), \( Y = 0.4 + 0.8i \) (complex solution)
4. \( X = 1.2 + 0.6i \), \( Y = 0.4 - 0.8i \) (complex solution)

If you're looking for real solutions, the valid ones are:

- \( X = 0.4 \), \( Y = -0.2 \)
- \( X = 2 \), \( Y = 1 \)

Would you like further details on how these solutions were derived or have any additional questions?

Here are 5 related questions for practice:
1. What happens if you change the constants on the right side of the equations?
2. How can you check if these solutions satisfy the original system of equations?
3. What methods can be used to solve systems of nonlinear equations?
4. Can you generalize this approach to systems of more than two variables?
5. What would happen if the equations involved higher powers of \( X \) and \( Y \)?

**Tip:** Always substitute your solutions back into the original equations to ensure they are correct!

Solve the system of equations: 5X(1 + 1/(X^2 + Y^2)) = 12 and 5Y(1 - 1/(X^2 + Y^2)) = 4.

Learn how to solve the nonlinear system of equations 5X(1 + 1/(X^2 + Y^2)) = 12 and 5Y(1 - 1/(X^2 + Y^2)) = 4 using algebraic techniques. This tutorial covers systems of nonlinear equations and provides detailed steps suitable for advanced high school or undergraduate students.

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