I am asked to solve the following system of equations using Cramer's Rule
$$
A^2 + B = x\\
AB = 1-x^2
$$
where $A$ and $B$ are 2 unknowns that are to be solved for some given $x$. ie I'm expecting the solution to be $A = a(x)$ and $B = b(x)$ where $a(x)$ and $b(x)$ is some function of $x$.
Is it possible to do so? Alternatively, can I differentiate both equations explicitly w.r.t. $x$ and apply Cramer's rule? I have a feeling this is the right way since this involves linear algebra techniques.
$$
2AA' + B' = 1\\
A'B + B'A = -2x
$$
UPDATE
Alright, guys…the answer is actually to differentiate the equations w.r.t. $x$. I do not know if this is meaningful to any one. Basically, both $A$ and $B$ are implicitly defined in $x$. So after applying the differentiation, which is above, the next step is to construct the matrix:
$$
\begin{pmatrix}
2A & 1 \\
B & A
\end{pmatrix}\begin{pmatrix}
A'\\B'
\end{pmatrix} = \begin{pmatrix}
1\\-2x
\end{pmatrix}
$$
Applying Cramer's Rule,
$$
\begin{align*}
A' &= \frac{A + 2B}{2A^2 – B}\\
\\
B' &= \frac{4Ax + B}{B – 2A^2}
\end{align*}
$$
Best Answer
Cramer's rule is for linear system of equations only. So, no, you can't apply it for the system above.
Update
Also, what's $x$ in your equations? Because if it's actually a variable, then your system is a functional equations. If it's just a number, never mind then.