Question

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

Answer 1

Given equations,

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

$x+(a-1)y=2\to \left(2\right)$

The given equation will have a solution if they intersect.

If two lines are not coincident and not parallel then they must intersect each other.

Coincident check

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

$a+1\ne \frac{a+1}{2}$

$\therefore$ Lines are not coincident. $foralla$

Parallel check:

Slopes must be equal

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

$a^2-1=-1\Rightarrow a=0$

If $a=0$, then there will be no solution.

$\forall$ $a=IR-\left\{0\right\}$ Solution will exist for given equation.