In this section I explain a few simple algebraic tricks that are not always explicitly taught in math classes for one reason or another.  The tricks can be used to simplify expressions or to make the expression look more complicated.

1. A simple shortcut

I discovered that many people have issues solving equations of the form

$$\displaystyle y = \frac{a}{x}$$

for the variable, $x$. Note that the equations can look much more complicated than this, but the general form is the same.  If you use the standard method taught in math classes throughout the United States, you would first  multiply both sides of the equation by $x$

$$\displaystyle yx = a$$

and then divide both sides by $y$ to get

$$\displaystyle x = \frac{a}{y}$$

The net effect is that the $x$ and $y$ exchanged places.  That's all there is to this shortcut! Simply exchanging the $x$ and the $y$ is much more efficient than performing two steps and you are less likely to make a mistake at some point. 

Example: Given

$$\displaystyle r^2 = \frac{3}{b^2} + \frac{a}{x^2-1}$$

Solve for x.

Solution:

$$\displaystyle r^2 - \frac{3}{b^2} = \frac{a}{x^2-1}$$

use the shortcut to get

$$\displaystyle x^2-1= \frac{a}{r^2 - 3/b^2}$$

then finish solving

$$\displaystyle x= \pm\sqrt{1 + \frac{a}{r^2 - 3/b^2}}$$

2. Multiplying by one (1)

A common way of manipulating mathematical expressions involves multiplying (or dividing) by the number one. Multiplying by one obviously does not change the value of an expression, but it can drastically change the appearance of the expression. There are many ways of writing the number one. Most fall into one of the categories below:

2.1 Special expressions for the number one

Suppose $y + 1 = \cos x$. Then

$$\displaystyle\frac{y+1}{\cos x} = 1$$

Also note

$$\displaystyle\frac{2}{2}=1$$

This demonstrates the most common type of expression for the number one. For example, if we start with the equation

$$\displaystyle \frac{y+1}{\cos x}y = 24 x$$

we can multiply by the first expression for one above to obtain

$$\displaystyle y^2+y = 24x\cos x$$

Then multiply by the second expression for one,

$$\displaystyle 2y^2+2y = 48x\cos x$$

The number one can also be expressed using other identities, for instance,

$$\displaystyle\cos^2x+\sin^2x = 1$$

and

$$\displaystyle -e^{i\pi} = 1$$

3. Adding zero

3.1. Add anything, then multiply it by zero