How do you write a system of equations?
Any set of equations is a system of equations. Whether it has solutions, let alone a unique solution is another question.
Playing with the question a little, given a solution in a number of variables, how do you create a set of equations to which they are the unique solution?
For example: Start with:
How do we generate a system of equations for which this is the only solution?
Option A: Do nothing. This is a set of equations with the desired unique solution. Downside: too obvious.
Option B: Since there are
This generally works well, except that we can create a set of equations that have multiple solutions. For example, if we chose
To avoid this, just try solving the question you create.
Option C: Go beyond linear. Plenty of scope for imagination here.
Downside: You will probably introduce additional unexpected irrational and/or complex solutions.