i could use some help in presenting a good solution for this problem :
Problem
Assume the system of equations in $x, y, z, t$
$ax+y+z+t=1$
$x+ay+z+t=b$
$x+y+az+t=b^2$
$x+y+z+at=b^3$
where $a, b$ are integer parameters. Determine the values of $a$ and $b$ for which the system has infinitely many solutions and the values of $a$ and $b$ for which the system has no solutions.
I have solved the equations and then for the values of $a$ that make the denominator equal to zero, i solved the numerators for $b$. I found the pairs of $a$ and $b$ that make the system impossible, or to have infinite solutions but i was wondering if there is a more "programming" way in SageMath. Thank you in advance.
Edit 1: I 've gone so far but now i can't manage to extract my last columns and create equations $= 0$ to solve as $b$.
my solution so far
I am trying to create the equations this way but it returns False, False, False.
What is the difference with that way of creating equations?