Thursday, October 13, 2011

How To Solve Equations

Knowing how to solve equations is a very important skill to have in mathematics courses.  There are all kinds of manipulations and substitutions that could be possible for any given equation, but knowing where and when to apply certain techniques is crucial to solving rational equations correctly.  In this post, I am going to go over several concepts that will be useful to you when it comes to solving rational equations.

To start, I will explain first-degree equations in one variable.  Quite simply, a first-degree equation is one in which there is only one variable.  In general, a first-degree or linear equation has the form:

ax + b = 0, where a and b are real numbers and a does not equal 0

I'm sure you are extremely familiar with this type of equation, though you may not know it by this name.  An example of a linear equation would be something like 3x - 12 = 0.  Undoubtedly, you can easily see that this equation is true when x = 4.  However, it is good to realize that the expression is neither true nor false until you substitute in a value for the variable.  Any value that makes the expression correct is called a solution or root of the equation.  To further classify this equation, 3x - 12 = 0 is also called a conditional equation, in that it is only true for certain values.

When you have two expressions that have the same solution (or root), these are called equivalent equations. Again, I'm sure you are familiar with the concept, but probably unfamiliar with this name.  When you have a first-degree equation in one variable, the general strategy that you typically employ is to express the equation equal to a series of equivalent equations, which you manipulate until you can reduce everything down to the solution to the equation.

The rules for generating equivalent equations are simple and intuitive.
  1. You can add or subtract the same value from both sides of the equation.  (A corollary to this is that you can add AND subtract the same value on one side, without changing the other... since adding x and then subtracting x means you really have done nothing!)
  2. You can multiply or divide each side of the equation by the same value.
  3. You can simplify one side of the equation without affecting the other side of the equation.
I think these rules are fairly self-explanatory, so I'm not going to bother going into any examples to demonstrate them.

When you have arrived at your solution / root of your equation, it is ALWAYS smart to take that value and substitute it into the original expression to verify that it is indeed true.  It always amazes me how many people arrive at incorrect answers and leave it at that, when a simple review and check can either tell you that you are correct, or your answer needs more work.  ALWAYS REMEMBER TO CHECK YOUR ANSWERS!

By checking your answers by substituting the solution into the equation, you sometimes will determine that the solution you have found CANNOT be true, in which case your solution is called an extraneous root.  An example of this would be where, when checking your solution, you determine that you have a 0 on the bottom of a fraction (the denominator).  A fraction with a zero in the denominator is undefined, and so you can conclude that the root you determined does not satisfy your equation.  Extraneous roots may develop especially if you use rule number 2 above, but you multiply both sides by an EXPRESSION rather than a single number.  (eg. you multiply both sides by (x + 2))

That is all I am going to say about how to solve equations for now, especially the first-order equations (or linear equations).  I will continue in my next post with a discussion of solving quadratic equations.

Sunday, October 9, 2011

Principal Axis Factoring

I'm just going to put a brief mention up here on principal axis factoring, a theoretical statistical method used to analyze common variance.  Principal axis factoring, also known as principal factor analysis or common factor analysis, is the most commonly used method in factor analysis.  According to Wikipedia, it "seeks the least number of factors which can account for the common variance (correlation) of a set of variables".  Factors are determined through the analysis of the common variance.

This concept of factor analysis is beyond the concepts that I would like to explain on my blog. It is much more of a university-level statistics concept than any of the other concepts that I have described so far.  If you would like a more thorough explanation of principal axis factoring, I highly suggest that you perform a Google search or find a good reference text book.  I mention it here only because of my recent posts on methods of factoring, and I was asked to specifically to reference this factoring method.

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