Copyright © University of Cambridge. All rights reserved.

Why do this problem?

Making substitutions to make the task in hand easier is an example of a valuable technique in many areas of mathematics (eg integration, transformations of functions, change of axes, diagonalisation of matrices etc.) By introducing students to this technique as an example of a general process it will help them to understand what is going on when they meet the process in many different mathematical situations. These equations give students useful practice in algebraic manipulation. They will need to look for symmmetrical features in the expressions and exploit the symmetry to make it easier to solve the equations.

When we use mathematical language to communicate ideas, a frame of reference is usually taken for granted unless we specify otherwise (eg base ten numbers). However the same mathematical relationships can be expressed with different frames of reference and some mathematical tasks are simpler in one frame of reference than in another.Typically 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.

Possible approach

It is helpful to introduce this problem with some discussion about switching frames of reference to make the equations easier to solve and how you are looking for symmetry in order to choose a good substitution.

One approach would be to divide the class into 4 groups and give each group one of the equations and ask them to discuss the symmetries they notice, to try possible substitutions and to solve their equation. You could then give assistance to the groups separately helping them as far as possible to do the work for themselves with your guidance rather than you as the teacher doing too much for them. This could lead into a homework. In the next lesson you could start with a class discussion of the symmetries in each equation in turn and then a representative from each group could present the solution to the class on the board.

Either before this lesson, or as part of the homework between your lessons, you might get the class to read the introduction to the article "The Why and How of Substitution" and perhaps to work through Example 1 in that article.

Possible extension

(1a) $\frac{x^2 - 10x + 15}{x^2 - 6x + 15}=\frac{3x}{x^2 - 8x + 15}.$

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

(2) $(2x - 3)^4 + (2x - 5)^4 = 2.$

(3a) $ (x - 2)(x + 1)(x + 4)(x + 7) = 19.$

(3b) $(12x - 1)(6x - 1)(4x - 1)(3x - 1) = 5.$

Possible support

Make up your own equation by taking a simple equation and making a substitution to make it more complicated. For example take any quadratic equation in $x$ and turn it into a quartic equation by substituting $x=X+\frac{1}{X}.$