Question

How do you write a system of equations?

Answer 1

Any set of equations is a system of equations. Whether it has solutions, let alone a unique solution is another question.

Explanation:

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.