Given this system of linear equations where ${a}$ is parameter.
\begin{align}ax + 18y &= 13a^2 - a \\5x + (a - 1)y &= 15 \end{align}
For which value of $a$ The system will have infinitely many solutions? Give me different ways to solve this problem. Need guide not solutions. I want to solve myself.
from the second equation we get $$x=3-\frac{1}{5}(a-1)y$$ plugging this in the first one $$a\left(3-\frac{1}{5}(a-1)y\right)+18y=13a^2-a$$ can you finish? from here we get $$y\left(18-\frac{1}{5}a(a-1)\right)=13a^2-4a$$ finally we obtain $$a^2-a-90\ne 0$$ and $$x=\frac{13a^3-14a^2+a-270}{a^2-a-90}$$ $$y=-\frac{5a(13a-4)}{a^2-a-90}$$