Solve for $x$ and $y$: $\frac{x}{a}=\frac{y}{b};\ ax+by=a^{2}+b^{2}$.

 Given: A pair of equations $\frac{x}{a}=\frac{y}{b};\ ax+by=a^{2}+b^{2}$.


To do: To solve the given pair of equations for $x$ and $y$.

Solution: 

Given equations:

$\frac{x}{a}=\frac{y}{b}$   

$\Rightarrow bx=ay$

$\Rightarrow bx-ay=0$       ........... $( 1)$

$ax+by=a^{2}+b^{2}$            ........... $( 2)$

On multiplying $( 1)$ by $a$ and $( 2)$ by $b$:

$abx-a^{2}y=0$                    ............ $( 3)$


$abx+b^{2}y=a^{2}b+b^{3}$  .......... $( 4)$

On subtracting $( 3)$ from $( 4)$:

$abx+b^{2}y-abx+a^{2}y=a^{2}b+b^{3}-0$

$\Rightarrow y( a^{2}+b^{2})=b( a^{2}+b^{2})$

$\Rightarrow y=b$, on substituting this value in $( 1)$

$bx-ab=0$

$\Rightarrow x=a$

Thus, $x=a$ & $y=b$

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Updated on: 10-Oct-2022

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