Solve the following system of equations by the method of cross-multiplication:

$\frac{x}{a}\ =\ \frac{y}{b}$
$ax\ +\ by\ =\ a^2\ +\ b^2$

Given: The system of equation given to us is $\frac{x}{a}\ =\ \frac{y}{b}$ ; $ax\ +\ by\ =\ a^2\ +\ b^2$

 To do: Solve the given system of equations by the method of cross-multiplication

Solution:  The system of equation can be written as

$\frac{x}{a}\ =\ \frac{y}{b}$ ---1)

$ax\ +\ by\ =\ a^2\ +\ b^2$---2)

Multiply equation 1) by b 

$\frac{bx}{a}\ =\ \frac{by}{b}$

$\frac{bx}{a}\ =y$ ---3)

Put the value of $y$  in 2)

$ax +b\frac{bx}{a} =a^2 +b^2$

$\frac{a^2x+b^2x}{a}=a^2+b^2$

$\frac{x(a^2+b^2)}{a}=a^2+b^2$

$x(a^2+b^2)=(a^2+b^2)a$

$x=\frac{(a^2+b^2)a}{a^2+b^2}$

$x=a$

Now put the value of $x$ in 3)

$\frac{bx}{a}\ =y$

$\frac{ba}{a}\ =y$

$y=b$

Therefore, the valueof $x$and $y$ is $a$ and $b$ respectively.

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Updated on: 10-Oct-2022

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