The pair of linear equations represented by these lines $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$

- If $\dfrac{a_{1}}{a_{2}} \neq \dfrac{b_{1}}{b_{2}}$ then the pair of linear equations has exactly one solution.
- If $\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} = \dfrac{c_{1}}{c_{2}} $ then the pair of linear equations has infinitely many solutions.
- If $\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} \neq \dfrac{c_{1}}{c_{2}} $then the pair of linear equations has no solution.

Given that, the equation $ax-(a+b)y=1$ and $(a-b)x+ay=5$

$\implies ax-(a+b)y-1 = 0,\;(a-b)x+ay – 5 = 0$

For unique solution$:\dfrac{a_{1}}{a_{2}} \neq \dfrac{b_{1}}{b_{2}} $

$\implies \dfrac{a}{a-b} \neq \dfrac{-(a+b)}{a}$

$\implies a^{2} + (a-b)(a+b) \neq 0$

$\implies a^{2} + (a^{2} – b^{2}) \neq 0$

$\implies 2a^{2} – b^{2} \neq 0$

$\implies 2a^{2} \neq b^{2}$

$\implies \dfrac{a^{2}}{b^{2}} \neq \dfrac{1}{2}$

$\implies a^{2} :b^{2} \neq 1:2$