Question

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

Answer 1

Given equations,

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

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

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

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

For solution to be defined, $1-a^2\ne 0$

$a\ne \pm 1$

$\left(1\right)\Rightarrow x=1-ay$

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

$\forall a=IR-\{-1,1\}$ , solution exists

Solution is $x=\frac{1-2a^2}{1-a^2},y=\frac{a}{1-a^2}$