<s>[INST] Solve three equations with three unknowns Solve the system:
$$\begin{cases}a+b+c=6\\ab+ac+bc=11\\abc=6\end{cases}$$
The solution is:
$a=1,b=2,c=3$
How can I solve it?
 [/INST] By method of substitution:
$$\begin{cases} b+c=6-a \\ a(6-a)+\frac{6}{a}=11 \\ bc=\frac{6}{a}\end{cases} \Rightarrow$$
$$a^3-6a^2+11a-6=0 \Rightarrow (a-1)(a-2)(a-3)=0 \Rightarrow a=1; 2; 3.$$
Substituting these and solving $$\begin{cases} b+c=6-a \\ bc=\frac{6}{a}\end{cases}$$
six solutions will be found:
$$(a,b,c)=(1,2,3); (1,3,2); (2,1,3); (2,3,1); (3,1,2); (3,2,1).$$
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