Question

For what values of $a$ does the following systems of equations have solutions? Find those solutions.
$x+ay=1,ax+y=$ $a^2$

Answer 1

Given equations,

$x+ay=1\to \left(1\right)$

$ax+y=a^2\to \left(2\right)$

$\left(1\right)\times a-\left(2\right)\Rightarrow (a^2-1)y=a-a^2$

$y=\frac{a-a^2}{a^2-1}$

The given equations will have solution if $a^2-1\ne 0$

$a\ne \pm 1$

$y=\frac{-a}{a+1}$

$\left(1\right)\Rightarrow x=1-ay=\frac{a^2+a+1}{a+1}$

Given equations will have solution if $a\ne \pm 1$

$x=\frac{x^2+a+1}{a+1},y=\frac{-a}{a+1}$ is solution of given equations $\forall a=IR-\{-1,1\}$