SOLVING EQUATIONS (General Information)
Anytime you see the word SOLVE, keep in mind that you are searching for the value of the indicated un-
known. (Or the value of it relative to the other variables in the equation.)
If you are Solving for x, for example: Then xmust be isolated on one side of the equation and all
other terms and factors must be correctly ”maneuvered” to the other side of the equation through correct
algebraic manipulation.
When completed: There should be no ”x’s” on the ”other side” of the equation. (If there were, you
would not have solved for x- It would be like giving a definition of ”table” and using the word ”table” in
your definition - it’s worthless.)
As we go through the semester: We will find new methods to solve equations - each time, the
method will bring us back to a simpler, more f amiliar equation that we should already know how to solve.
Thus, it is very important to look for PATTERNS or TYPES of equations so that you will know the most
efficient method to apply as you see them.
Steps in Solving: When solving equations, each step should be an equation that is ”equivalent” to
the previous equation. (See E xceptions Below) This may sound like double talk, but for two equations to
be equivalent, they must have the same solution(s). The solution ”is the value of the unknown that makes
the given equation a true statement.” (For example: If x+ 5 = 12, then x= 7 is the solution or root because
7 + 5 = 12 is true). There are theorems or rules that tell us that we can: ”add, subtract, multiply, divide”
both sides of an equation by the same number (as long as we don’t multiply or divide by 0), and the resulting
equation will be ”equivalent” to the previous one. These basic ”algebraic maneuvers” allow us to solve all
linear (1st degree) equations and some rational (fraction) equations.
Non-linear Equations & the Multiplicative Property of Zero:Remember if you have an equation
that is ”non-linear”, that is it is of degree greater than one, you should do the following:
First: Set the Equation Equal to 0.
Second: FACTOR.
Third: Set Each Factor Equal to 0. This last step, enabling us to solve only simple linear equations
is possible because of the Multiplicative Property of Zer o which says that the ONLY way that product of
factors could be zero is that one of them has to equal 0.
Exceptions: These will be discussed later in the semester; but this is the basic idea: As the equa-
tion is stated, it can not be solved by the means that we have of keeping equations equivalent. We have to
employ a method that keeps all of the original solutions, but it may also introduce more solutions. That is,
as we are going through our steps, instead of every equation being equivalent (having the same solutions),
we produce a step that has an equation that could have extra or extraneous solutions. Obviously, it is good
that we did not lose any solutions or this would have been a very poor method. The extra work caused by
the possibility of extra solutions is that we will be required to CHECK our ANSWERS!
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