# How do you write a system of equations?

Jul 12, 2015

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?

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

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:

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

This generally works well, except that we can create a set of equations that have multiple solutions. For example, if we chose $3 a - 3 b - c = 2$ for the third equation instead of $2 a - 3 b - 2 c = 5$ 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:

$3 a + 2 b + c = 7$
$3 {a}^{2} + 2 {b}^{2} + {c}^{2} = 55$
$3 {a}^{3} + 2 {b}^{3} + {c}^{3} = 203$

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