"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 . We make the substitution , and we now have a new expression, equation or graph involving the given terms, the variable and the function . Since we are free to choose to be any function we like, it is highly likely that for a suitable choice of the new expression in will be simpler than the original expression in . The skill lies in the selection of ; the rest is just the algebraic manipulation of the variables.
Let us now look at some examples with these ideas in mind.
Example Polynomial Equations
Let us consider the polynomial equation
If we expand the left hand side we get a quartic in which we cannot solve. However, we notice that the left hand side has a certain symmetry, namely . The roots of the left hand side are symmetric about the value , and this suggests that we should make a substitution that exploits this fact. Let us try ; that is, we change the variable so that the symmetry is now about the origin (after all, and looks better than and ). With this we have
or . This is a quadratic equation in which we can solve to give two values of and four values of corresponding to the four solutions to the original equation. However, we can also simplify it with another substitution. The numbers and are symmetric about so we now make the substitution . This gives so that
You should check that if we put in this formula we do get the expected solutions . What do you get if ?
Example Rational functions
We want to solve the following equation:
By clearing fractions this becomes a quartic equation which is difficult to solve. Observing the occurrences of , and the symmetry of , and , we can turn this into a quadratic equation by substituting . We get the equation
This simplifies to , so that is or . Each value of gives a quadratic equation in , giving four solutions of the original equation in . The two quadratic equations are
These equations simplify to
and the four solutions are
Example Integration by substitution
Here a trigonometric substitution leads to a simpler integral. Because of the relation , we substitute and and get
To return to an expression in terms of we use , and the integral we want is
Example Area inside an ellipse
In order to find the the area inside the ellipse , we can use the transformation to change the ellipse into a circle. Since the lengths in the --direction are changed by a factor , and the lengths in the --direction remain the same, the area is changed by a factor . Thus
which gives the area of the ellipse as , that is .
Consider a general polynomial
where here "'' means powers of of order or less. If we now choose we see that
in other words, by changing the variable we can remove the term of degree . While the effect of this substitution may not seem spectacular, it is important. It is exactly 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 in a polynomial equation of degree is minus the sum of the roots of the equation so this substitution is such that the chosen value of is the average value of the roots of the polynomial.
Example Transformations of the plane
In Example we showed how to remove the term in from a polynomial of degree . Now we are going to show how, given the equation of a conic, for example,
we can remove the term and so more easily discover the properties of the conic. First, if we make the substitution
Thus equation gives an ellipse if , a hyperbola if , and it reduces to a pair of lines if . The question, however, is (as at the start of this article) "How could I have seen this?" We are going to change the variables to new variables by rotating the plane by an angle . As we do not yet know which value to take, we work with a general and make this choice later. A rotation of the plane by an angle is given by
If we substitute these in equation we obtain
and so if we now choose so that , we see that the term will vanish. Thus we take , and this with gives the values of and as in and hence the equation of the conic as in .
More generally, if we have an equation