Cambridge O Level Additional Mathematics · Syllabus 4037 · Simultaneous Equations
Symmetric System
What is Symmetric System?
A pair of simultaneous equations that 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 O Level 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: \(x-y=\pm2\).
Questions students ask about Symmetric System
How do you solve a symmetric system given \(x^2+y^2\) and \(xy\)?
Use the identities \((x+y)^2=x^2+2xy+y^2\) and \((x-y)^2=x^2-2xy+y^2\) to turn the given values into \(x+y\) and \(x-y\), then combine them to find \(x\) and \(y\). Write \(\pm\) the moment you take a square root, because \((x-y)^2=4\) gives \(x-y=\pm2\), not just \(2\). A symmetric system has mirror-image solutions, so an answer with an odd number of pairs deserves a second look.