{(ax+by=3),(bx+ay=3):} (a^2 -b^2 is not 0)

The system of equations is:

\[
1. \, ax + by = 3 
\]
\[
2. \, bx + ay = 3
\]

We are also given the condition \( a^2 - b^2 \neq 0 \), which implies that \( a \neq b \) and \( a \neq -b \).

### Step-by-step solution:

1. **Add the two equations**:

   \[
   (ax + by) + (bx + ay) = 3 + 3
   \]

   This simplifies to:

   \[
   (a + b)x + (a + b)y = 6
   \]

   Factor out the common term:

   \[
   (a + b)(x + y) = 6 \tag{3}
   \]

2. **Subtract the second equation from the first**:

   \[
   (ax + by) - (bx + ay) = 3 - 3
   \]

   This simplifies to:

   \[
   (a - b)x + (b - a)y = 0
   \]

   Rearranging gives:

   \[
   (a - b)(x - y) = 0 \tag{4}
   \]

### Conclusion from equations (3) and (4):

- From equation (4), we have two cases:
  1. If \( a \neq b \), then \( x = y \).
  2. If \( a = b \), this contradicts the given condition \( a^2 - b^2 \neq 0 \).

- Now, using \( x = y \) in equation (3), we get:

   \[
   (a + b)(x + x) = 6
   \]
   \[
   (a + b)(2x) = 6
   \]
   \[
   2(a + b)x = 6
   \]
   \[
   x = \frac{3}{a + b}
   \]

Since \( x = y \), we also have:

\[
y = \frac{3}{a + b}
\]

### Final Answer:
The solution to the system is \( x = y = \frac{3}{a + b} \).

Let me know if you'd like more details on any of these steps!

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### Further exploration:

1. What happens if \( a = 1 \) and \( b = 2 \)?
2. How do we handle systems where \( a^2 - b^2 = 0 \)?
3. What is the geometric interpretation of this system of equations?
4. How does the solution change if the right-hand side constants were different?
5. Can we solve similar systems with three or more equations?

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**Tip**: When dealing with systems of equations, adding and subtracting equations is a powerful technique to eliminate variables!

Given the system of equations {(ax+by=3),(bx+ay=3):} where a^2 - b^2 is not 0, solve for x and y.

Learn how to solve the system of linear equations ax + by = 3 and bx + ay = 3 with detailed steps. This problem explores algebraic techniques such as equation addition, subtraction, and properties of simultaneous equations. Suitable for students in Grades 10-12.

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