Summary: Algebra

When faced with an algebraic expression or equation, there are only two types of things you can do to it without changing the quantitative relationship that it describes.

Re-write one or more terms in an equivalent form

This can be done to any expression (no equal sign) or equation (with an equal sign) at any time.

There are three common ways in which this is done: by combining like terms, expanding terms, and doing something that cancels itself out.

Combining like terms:

2x-5+x+1\\*~\\*=~2x+x-5+1~~~~~~~\text{commutative property of addition}\\*~\\* =~(2+1)(x)-5+1~~\text{factoring}\\*~\\*=~3x-4~~~~~~~~~~~~~~~~~\text{combining like terms}

Expanding terms by substituting an equivalent expression for a term:


Doing something that cancels itself out or has no net effect, like adding zero, adding then subtracting the same number, or multiplying then dividing by the same number:

3x -4\\*~\\*=~3x-4+7-7\\*~\\*=~\dfrac{5(3x-4)}{5}+7-7

Do exactly the same thing to both sides

When working with an equation, you must always preserve the “balance” of the expressions on both sides of the equation. All of the steps mentioned in the previous section will do this. And there is one additional step you can take when working with an equation: you may do exactly the same thing to the entirety of each side of the equation.

In other words, treat each entire side as a single entity (by putting parentheses around it), and do the same thing to both (add a number to both, square both, take the same function of both, etc.). For example:

2x-5~=~4\\*~\\*(2x-5)^3~=~(4)^3 ~~\text{cube both sides}

Applying the same function or operation to the entirety of each side will preserve the relationship that the equality describes. Some functions, like squaring, should be used with caution since they can introduce phantom solutions to the original problem:

x~=~-3\\*~\\*x^2~=~9~~~\text{square both sides}\\*~\\*x~=~\pm 3~~\text{take the square root of both sides}

Note that positive 3 was not a solution that I started with, but was introduced as a possible solution by the squaring of both sides… which caused sign information to be lost.

Now go manipulate some expressions

That’s it. Algebra is the art of changing the way an expression looks, without changing the quantitative relationship that it describes. It mostly boils down to the above actions, carried out as many times as necessary to achieve your objective. The “art” of it is learning to be efficient in your manipulations. The more experience you acquire manipulating expressions and equations, particularly “ugly” ones, the more easily you will see the possibilities when faced with a new problem to work on.

By Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.

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