X/a+y/b=2 ax-by=a^2-b^2 Solve with linear euarion in two variables

$\begin{array}{l}\text{Given:}\\ \frac{x}{a}+\frac{y}{b}=2......\left(1\right);\ \ \ \ ax-by=a^2-b^2......\left(2\right)\\ \\ \frac{bx+ay}{ab}=2\\ \left[bx+ay=2ab\right]\times a.......\left(3\right)\\ \left[ax-by=a^2-b^2\right]\times b.......\left(4\right)\\ \text{Eliminating eq(4) from eq(3)}\\ abx+a^2y=2a^2b\\ bx-b^2y=a^2b-b^3\\ -\ \ \ +\ \ \ \ \ \ \ \ -\\ a^2y+b^2y=2a^2b-(a^2b-b^3)\\ y(a^2+b^2)=2a^b-a^2b+b^3\\ y(a^2+b^2)=a^2b+b^3\\ y(a^2+b^2)=b(a^2+b^2)\\ y=b.......\left(5\right)\\ \text{Putting eq(5) in eq(3)}\\ bx+ay=2ab\ \\ bx+a(b)=2ab\ \\ \Rightarrow bx=2ab-ab\ \\ \Rightarrow bx=ab\ \\ \Rightarrow\ x=\frac{ab}{b}\\ \Rightarrow x=a\\ So,\ x=a\ and\ y=b\\ \end{array}$

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