Standard Tricks

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

2.1 Special expressions for the number one

(dividing one equation by another)

h

3. Adding zero

3.1. Add anything, then multiply it by zero