Question

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

Answer 1

Given equations,

$ax+y=a\to \left(1\right)$

$x+ay=a^2\to \left(2\right)$

For existence of solution, both lines must intersect each other.

If lines are coincident and not parallel then only they intersect with each other.

Coincidence check: $\frac{a}{1}=\frac{1}{a}=\frac{a}{a^2}$

$a=\frac{1}{a}=\frac{1}{a}$

$a^2=1$

$a=\pm 1$

If $a=\pm 1$ , then equations won't finite solutions.

Parallel check: $-a=\frac{-1}{a}$

$a^2=1$

$a=\pm 1$ (same condition)

$\therefore$ Finite solution exists if $a=IR-\{-1,1\}$

$\left(1\right)\times a-\left(2\right)\Rightarrow (a^2-1)x=0$

$x=0$

$y=a$

$x=0,y=a$ is solution of given equations $\forall a=IR-\{-1,1\}$