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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 9-10.
A system is symmetric when it remains the same after permuting the unknowns.
Notice how, by interchanging the unknowns, the system remains the same (the names of the unknowns do not matter).
Such a system is easily solved by adding all equations:
Non-linear systems are a lot more tricky.
Example: The system is quadratic since the unknowns come up in products of two. Therefore, we expect to obtain a maximum of two solutions:
Example: This is a symmetric cubic system (the highest power of a term is 3). Therefore, we expect to have a maximum of three distinct solutions:
which leads to the idea of using a different set of unknowns (change of variable):
The system becomes:
with the obvious solution:
However, what if the examiner offers only the following answer choices:
Remember! we were expecting three solutions. The obvious solution is not the only one.
Since plugging in all the horrible expressions would take a long time and be prone to error, it is more reliable to solve the equation:
notice that one of the roots matches an answer.
Bottom line if the system is symmetric, a change of variable:
is always possible and might be useful in simplifying the problem.