##### Stage: 5

Published October 2010,February 2011.

Introduction

"How could I have seen that?" This is a common response to seeing a

substitution in mathematics, and this article attempts to answer this

question. Sadly, the technique of substitution is often presented

without mentioning the general idea behind all substitutions. The effective use of substitution depends on two things: first, given a situation in which

variables occur, a substitution is nothing more than a change of variable; second, it is only effective if the change of variable simplifies the situation and, hopefully, enables one to solve the simplified problem.

There is no easy route to this: substitution will only work if the the

original situation has some kind of symmetry or special property that we

can exploit, and the skill in using the method of substitution depends

on noticing this. Thus we should always be looking for special features in the problem, and then be prepared

to change the variable(s) to exploit these features. Of course,

once we have solved the problem in the new variables we have to rewrite

the solution in terms of the original variables.

The main idea behind substitution, then, is this. We are given some expression, or equation or graph involving the variable

Let us now look at some examples with these ideas in mind.

Example

Let us consider the polynomial equation

If we expand the left hand side we get a quartic in

origin (after all,

or

corresponding to the four solutions to the original equation. However,

we can also simplify it with another substitution. The numbers

so that

You should check that if we put

Example

We want to solve the following equation:

By clearing fractions this becomes a quartic equation which is difficult to solve. Observing the occurrences of

This simplifies to

These equations simplify to

and the four solutions are

Example

Evaluate

Here a trigonometric substitution leads to a simpler integral. Because of the relation

To return to an expression in terms of

Example

In order to find the the area inside the ellipse

which gives the area of the ellipse as

Example

Consider a general polynomial

where here "

in other words, by changing the variable we can remove the term of degree

what we do when we 'complete the square' to solve quadratic equations,

and this is the method used to find the formula for the roots of a

quadratic equation. It is also the

first step in solving cubic equations, for there it says that we only

need consider equations of the form

Finally, it is worth noting that the coefficient of

Example

In Example

we can remove the

Thus equation

The question, however, is (as at the start of this article) "How could I

have seen this?" We are going to change the variables

general

If we substitute these in equation

and so if we now choose

More generally, if we have an equation

and then apply the method given above. In this way, by a combination of

a translation and a rotation, we can change the variables so that the

conic

as the axes of symmetry of the conic.

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