If the following equations are consistent and have more than one solution, what is the value of $a$?
Given
$u+v=-(av+1)$
$u+2v=-a(v-1)$
$3u+8v=a+2$
I was thinking that system of equation is inconsistent then $(u,v)$ obtained by solving two equations must satisfy the third. But then the statement "have more than one solution" does not come into picture. Could someone suggest a better approach here that takes everything into account.
Make sure the rank of the system's matrix is $1$ (this means the system's variables are $u,v$ and all minors of rank $2$ must be $0$). If the rank is $2$ you'll get one solution at best.