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Algebra - Notes
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## Linear Equations

We’ll start off the solving portion of this chapter by solving linear equations.  A linear equation is any equation that can be written in the form

where a and b are real numbers and x is a variable.  This form is sometimes called the standard form of a linear equation.  Note that most linear equations will not start off in this form.  Also, the variable may or may not be an x so don’t get too locked into always seeing an x there.

To solve linear equations we will make heavy use of the following facts.

1. If
then
for any c.  All this is saying is that we can add a number, c, to both sides of the equation and not change the equation.

2. If
then
for any c.  As with the last property we can subtract a number, c, from both sides of an equation.

3. If
then
for any c.  Like addition and subtraction we can multiply both sides of an equation by a number, c, without changing the equation.

4. If
then
for any non-zero c.  We can divide both sides of an equation by a non-zero number, c, without changing the equation.

These facts form the basis of almost all the solving techniques that we’ll be looking at in this chapter so it’s very important that you know them and don’t forget about them.  One way to think of these rules is the following.  What we do to one side of an equation we have to do to the other side of the equation.  If you remember that then you will always get these facts correct.

In this section we will be solving linear equations and there is a nice simple process for solving linear equations.  Let’s first summarize the process and then we will work some examples.

Process for Solving Linear Equations

 If the equation contains any fractions use the least common denominator to clear the fractions.  We will do this by multiplying both sides of the equation by the LCD.  Also, if there are variables in the denominators of the fractions identify values of the variable which will give division by zero as we will need to avoid these values in our solution. Simplify both sides of the equation.  This means clearing out any parenthesis, and combining like terms. Use the first two facts above to get all terms with the variable in them on one side of the equations (combining into a single term of course) and all constants on the other side. If the coefficient of the variable is not a one use the third or fourth fact above (this will depend on just what the number is) to make the coefficient a one.  Note that we usually just divide both sides of the equation by the coefficient if it is an integer or multiply both sides of the equation by the reciprocal of the coefficient if it is a fraction. VERIFY YOUR ANSWER!  This is the final step and the most often skipped step, yet it is probably the most important step in the process.  With this step you can know whether or not you got the correct answer long before your instructor ever looks at it.  We verify the answer by plugging the results from the previous steps into the original equation.  It is very important to plug into the original equation since you may have made a mistake in the very first step that led you to an incorrect answer. Also, if there were fractions in the problem and there were values of the variable that give division by zero (recall the first step…) it is important to make sure that one of these values didn’t end up in the solution set.  It is possible, as we’ll see in an example, to have these values show up in the solution set.

Let’s take a look at some examples.

Okay, in the last couple of parts of the previous example we kept going on about watching out for division by zero problems and yet we never did get a solution where that was an issue.  So, we should now do a couple of those problems to see how they work.

 Example 2  Solve each of the following equations. (a)    [Solution] (b)    [Solution]   Solution (a)   The first step is to factor the denominators to get the LCD.                                                          So, the LCD is  and we will need to avoid  and  so we don’t get division by zero.   Here is the work for this problem.                                  So, we get a “solution” that is in the list of numbers that we need to avoid so we don’t get division by zero and so we can’t use it as a solution.  However, this is also the only possible solution.  That is okay.  This just means that this equation has no solution.   (b)   The LCD for this equation is  and we will need to avoid  so we don’t get division by zero.  Here is the work for this equation.                                                   So, we once again arrive at the single value of x that we needed to avoid so we didn’t get division by zero.  Therefore, this equation has no solution.

So, as we’ve seen we do need to be careful with division by zero issues when we start off with equations that contain rational expressions.

At this point we should probably also acknowledge that provided we don’t have any division by zero issues (such as those in the last set of examples) linear equations will have exactly one solution.  We will never get more than one solution and the only time that we won’t get any solutions is if we run across a division by zero problems with the “solution”.

Before leaving this section we should note that many of the techniques for solving linear equations will show up time and again as we cover different kinds of equations so it very important that you understand this process.

 Solutions and Solution Sets Previous Section Next Section Applications of Linear Equations Preliminaries Previous Chapter Next Chapter Graphing and Functions

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