Technique

Complete each equation to the common sum

Algebra • Equations • Symmetric Equations

Statement

In a cyclic system where each equation involves every variable but one, add the missing variable to both sides. Each right-hand side then becomes the full symmetric sum

s = x + y + z

, which is common to all the equations, so every variable satisfies one and the same relation in terms of

s

How to use it

1Use it whenever the right-hand sides read 'all the variables except this one'.
2After completing, factor the left side, e.g. $x^{2} + x = x (x + 1)$ , to get a relation you can divide by.
3A symmetric expression can then be rewritten with $s$ as a common denominator, after which $s$ cancels.

Worked example

x^{2} - 5 = y + z

y^{2} - 5 = z + x

z^{2} - 5 = x + y

and

x + y + z = 9

, find

x (x + 1)

Step-by-step

1
$x^{2} + x - 5 = x + y + z$
Why: Add $x$ to both sides so the right side is the full sum.
2
$x^{2} + x - 5 = 9$
Why: The common sum is given.
3
$x (x + 1) = 14$

Answer

14

Watch out

Dividing by

s

later requires

s \neq = 0

; a system of this shape usually also admits the all-zero solution, for which the conclusion fails.

Also known as

add the missing variable, complete the symmetric sum, cyclic system common total

Complete each equation to the common sum

Statement

Worked example

Appears in past papers

2019 (I)

2018 (II)

2017 (II)

2016 (II)