<s>[INST] Can the system of equations be extracted from its solution? While I was solving the secondary school exam of 2014 I came across a question that states:
After solving those equations: $a_{1}x + b_{1}y = c_{1}$ and $a_{2}x + b_{2}y = c_{2}$, we found that x = $\frac{-7}{\begin{vmatrix}
  3 & 1 \\
  1 & -2
 \end{vmatrix}}$ and y = $\frac{-21}{\begin{vmatrix}
  3 & 1 \\
  1 & -2
 \end{vmatrix}}$, then $c_{1} = .....$ and $c_{2} = .....$
The answer is $c_{1} = 6$ and $c_{2} = -5$ in the ministry's model answers on its website with no steps available ,but it was probably solved by Cramer's rule.
Is this answer right and does $c_{1}$ and $c_{2}$ have only one value?
 [/INST] Presumably the denominator $\left|\matrix{3 & 1\cr 1 & -2}\right|$ indicates that the coefficient matrix $\pmatrix{a_1 & b_1\cr a_2 & b_2} = \pmatrix{3 & 1\cr 1 & -2}$.  Then we have
$$ \pmatrix{c_1 \cr c_2} = \pmatrix{3 & 1\cr 1 & -2} \pmatrix{x\cr y\cr} = \pmatrix{3 & 1\cr 1 & -2} \pmatrix{1\cr 3\cr} = \pmatrix{6\cr -5\cr}$$
agreeing with the ministry's answer.
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