Choose a coefficient so that the system of equations has exactly one solution Choose the value of $a$ from $\{-4, 88\}$ so that the system of equations has exactly one solution:
$$2x +20y+3z=1$$
$$2x +2y+41z=2$$
$$ax -22y-44z=-3$$
I tried solving the system for $a=-4$ and after few steps found out that it was inconsistent. I would like to ask whether there is a faster way to determine the solution (so that I don't have to plug the numbers in). Thanks.

By Cramer's Rule,
$A$ =
$\begin{pmatrix}
2&20&3\\
2&2&41\\
a&-22&-44
\end{pmatrix}$
;
$A_x=
\begin{pmatrix}
1&20&3\\
2&2&41\\
-3&-22&-44
\end{pmatrix}$
;
$A_y=$
$\begin{pmatrix}
2&1&3\\
2&2&41\\
a&-3&-44
\end{pmatrix}$
;
$A_z=$
$\begin{pmatrix}
2&20&1\\
2&2&2\\
a&-22&-3
\end{pmatrix}$
;
When $A$ is non-zero, the equations are linearly independent, so $x=A_x/A$ ....can be uniquely determined.
