How do you write a system of equations?

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:

#a = 2#
#b = 3#
#c = -5#

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 #3# unknowns, write #3# linear equations with varying small integer multipliers for the unknowns:

#a + b + c = 0#
#a - 2b - c = 1#
#2a - 3b -2c = 5#

This generally works well, except that we can create a set of equations that have multiple solutions. For example, if we chose #3a - 3b - c = 2# for the third equation instead of #2a - 3b - 2c = 5# then there would be an infinite number of solutions.

To avoid this, just try solving the question you create.

Option C: Go beyond linear. Plenty of scope for imagination here.

For example:

#3a + 2b + c = 7#
#3a^2 + 2b^2 + c^2 = 55#
#3a^3 + 2b^3 + c^3 = 203#

Downside: You will probably introduce additional unexpected irrational and/or complex solutions.