Sadly, most people learn mathematics by thinking of it as a bunch of recipes to handle certain problems. Actually mathematics is a web of facts, concepts, and logical reasoning, and a vastly more efficient approach to understanding it is to focus on the principles and connections rather than on isolated facts.
Isaac Newton, considered by many the greatest scientist that ever lived, and certainly one of the greatest mathematicians that ever lived, put it this way in a letter to Nathaniel Hawes on 25 May 1694:
You want to be one who reasons nimbly and efficiently, rather than a vulgar mechanic!
Sprinkled through these pages are explicit descriptions of some of
the principles that can be used in building and understanding not just the curriculum of
Intermediate Algebra, but all of mathematics. They are typeset in green for better visibility,
and they are all listed on this particular page for your reference.
Some are much more profound than others, but all of them will empower
you tremendously if you consciously apply them and if you actively
look for them and observe them in action as you go through this
course.
This principle is used ubiquitously in mathematics. It has a major
consequence: in order to understand a piece of mathematics you have
to understand what preceded it. Following is a more elaborate version
of the same principle:
Introduce concepts in a simple context and
then generalize them in such a way that rules and facts that are true
in the simple context remain true in the more general
context.
You can see this principle in action for example in the way we build
the number system or define powers with exponents other than natural numbers.
Before you attempt the solution of a problem or start studying a
subject, it pays to think explicitly about what you expect your
solutions to look like, or what you expect to learn. There are two
possibilities: your expectations are met, or they are not. In
the first case you feel reassured and on top of things which is nice.
In the second case there are two possibilities. Either you made a
mistake and now that you are alert to the fact you can recover from
it. Or, and this is the most exciting possibility, your expectations
were based on some misunderstanding and now you have a chance to
improve your understanding and learn something new.
Applying the same operation on
both sides of a valid equation gives another valid
equation.
Think of an equation as one of those old fashioned scales where you
match an unknown weight on one side with a collection known weights on
the other. The weights on the two sides are in balance. If you do
the same thing to the weight on each side they will still be in balance.
To solve an equation figure out what bothers
you and then apply a suitable operation on both sides of the equation to get
rid of it.
This is the only way to solve an equation. You don't need to memorize
techniques for example for each item in the following list of problems taken from
our textbook: linear equations in standard form, linear equations in
non-standard form, linear equations involving decimals, linear
equations containing fractions, linear equations: special cases,
linear equations using ratios, linear equations involving proportions,
linear equations in percent problems. There is a similar litany for
quadratic equations, and one for radical equations. Don't believe any
of this, just understand how to manipulate algebraic expressions and
use common sense and the above principle.
Building Mathematics
Reduce your problem to one you have solved before.
Expectations
Always Have Expectations
Solving Equations