Solve for x and y: a^2 x+b^2 y=c^2, b^2 x+a^2 y=d^2 - Vidyadip

Question

Solve for $x$ and $y$:
$a^2 x+b^2 y=c^2,$
$ b^2 x+a^2 y=d^2$

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Answer

$a^2 x+b^2 y=c^2,...(i)$
$b^2 x+a^2 y=d^2 ...(ii)$
Multiplying $(i)$ by $a^2$ and $(ii)$ by $b^2$ and subtracting, we get
$\Rightarrow\text{a}^4\text{x} - \text{b}^4\text{x} = \text{a}^2\text{b}^2 - \text{b}^2\text{d}^2$
$\Rightarrow\text{x}=\frac{\text{a}^2\text{c}^2-\text{b}^2\text{d}^2}{\text{a}^4-\text{b}^4}$
Multiplying $(i)$ by $b^2$ and $(ii)$ by $a^2$ and subtracting, we get
$\Rightarrow\text{b}^4\text{y} - \text{a}^4\text{y} = \text{b}^2\text{c}^2 - \text{a}^2\text{d}^2$
$\Rightarrow\text{y}=\frac{\text{b}^2\text{c}^2-\text{a}^2\text{d}^2}{\text{b}^4-\text{a}^4}$
So, $\text{x}=\frac{\text{a}^2\text{c}^2-\text{b}^2\text{d}^2}{\text{a}^4-\text{b}^4}$ and $\text{y}=\frac{\text{b}^2\text{c}^2-\text{a}^2\text{d}^2}{\text{b}^4-\text{a}^4}$

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