Algebraic Equation Solving: Finding the Values of X and Y

Problem Statement

• Given equation: x^2 - y^2 = (x - y)^2
• Objective: Find the values of X and Y.

Solution Steps

1. Expand the right-hand side of the equation

   • Use the identity: (a - b)^2 = a^2 - 2ab + b^2
   • Expanded form: x^2 - y^2 = x^2 - 2xy + y^2

2. Subtract $x^2$ from both sides to simplify

   • Resulting equation: -y^2 = -2xy + y^2

3. Rearrange terms to one side to set the equation to zero

   • Final form: 2xy - 2y^2 = 0

4. Factor out common terms

   • Taking out 2y, we get: 2y(x - y) = 0

5. Find solutions for X and Y

   • Equate each factor to zero: 2y = 0 and x - y = 0
   • Solutions:
     • y = 0
     • x = y

Verification of Solutions

1. For y = 0:

   • Substitute in original equation: x^2 - 0^2 = (x - 0)^2
   • Result: x^2 = x^2 (Valid)

2. For x = y:

   • Substitute in original equation: y^2 - y^2 = (y - y)^2
   • Result: 0 = 0 (Valid)

Conclusion

• The valid solutions for the given equation are y = 0 and x = y.

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