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## 'Agile Algebra' printed from http://nrich.maths.org/

The following equations are difficult to solve by direct attack but
if you look for symmetric features and make simple substitutions
they become much easier to solve.

(1) $\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6.$

(2) $x^4 -8x^3 + 17x^2 -8x + 1 = 0.$

(3) $(x-4)(x-5)(x-6)(x-7) = 1680.$

(4) $(8x+7)^2(4x+3)(x+1)=\frac{9}{2}.$

NOTES AND BACKGROUND

This is an example of a process which occurs frequently in
mathematics. Let's refer to two frames of reference as A and B and
say we have a problem stated in A, then the technique is to map the
given relationships to B, work in B and then map the results back
to A. All these equations have symmetry of one sort or another. By
using the symmetry to make a substitution you can change the
variable and get an equation which is easier to solve. After that
you have to use the solutions you have found and go back to find
the corresponding solutions of the original equation.

To find out more about this general technique see the article

"The Why and How of
Substitution".