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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.

Example:

equation

equation

equation

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:

equation

equation

equation

equation

equation

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:

equation

equation

equation

equation

equation

equation

equation

equation

equation

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:

equation

equation

Notice that:

equation

which leads to the idea of using a different set of unknowns (change of variable):

equation

The system becomes:

equation

equation

with the obvious solution:

equation

equation

However, what if the examiner offers only the following answer choices:

equation

equation

equation

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:

equation

equation

equation

equation

equation

notice that one of the roots matches an answer.

Bottom line if the system is symmetric, a change of variable:

equation

is always possible and might be useful in simplifying the problem.