Cambridge IGCSE Additional Mathematics · Syllabus 0606 · Simultaneous Equations
Symmetric System
What is Symmetric System?
A pair of simultaneous equations in which each equation is unchanged when the two unknowns are interchanged, so that every expression in it is built from the sum x + y and the product xy. Such a system is solved most efficiently through the identities (x + y) squared equals x squared plus y squared plus 2xy and (x - y) squared equals x squared plus y squared minus 2xy, which convert the given information into the sum and the difference of the unknowns; because the solutions come in interchanged pairs, each sign combination produced must be tested rather than assumed.
This definition is part of the Simultaneous Equations chapter in Cambridge IGCSE Additional Mathematics.
Common mistakes with Symmetric System
- “\((x-y)^2=4\), so \(x-y=2\).” WHY IT FAILS Every positive number has two square roots. Taking only the positive branch halves the solution set, and in a symmetric system that means losing two of the four ordered pairs. FIX Write \(\pm\) at the moment you take the root, not afterwards: \(x-y=\pm2\).