Question

Solve for x and y:
$$\begin{gathered} \frac{x}{a}+\frac{y}{b}=a+b \\ \frac{x}{a^2}+\frac{y}{b^2}=2 \end{gathered}$$

Answer 1

The given equations are
$$\begin{gathered} \frac{x}{a}+\frac{y}{b}=a+b.....\left(i\right) \\ \frac{x}{a^2}+\frac{y}{b^2}=2.....\left(ii\right) \end{gathered}$$
Multiplying (i) by b and (ii) by b² and subtracting, we get
$$\begin{gathered} \frac{bx}{a}-\frac{b^{2}x}{a^2}=ab+b^2-2b^2 \\ \Rightarrow \frac{ab-b^2}{a^2}x=ab-b^2 \\ \Rightarrow x=\frac{\left(ab-b^2\right)a^2}{ab-b^2}=a^2 \end{gathered}$$
Now, substituting x = a² in (i), we get
$$\begin{gathered} \frac{a^2}{a}+\frac{y}{b}=a+b \\ \Rightarrow \frac{y}{b}=a+b-a=b \\ \Rightarrow y=b^2 \end{gathered}$$

Hence, $x=a^2\mathrm{and}y=b^2$.