Linear Equations with One Variable  
 
An equation involving one variable (x) of the general form ax + b = 0 is called a linear equation with one variable. As will be shown, a linear equation is called linear simply because it pertains to a line. It is also called an equation of the first degree, because the exponent of the unknown x is 1.

Why do we need to study this type of linear equations? Let's look at an example.

Assume a test consists of 25 multiple-choice questions. Each correct answer awards a student 4 points while no answer or a false answer penalizes a student 1 point. If a student obtains 80 points, how many questions did he answer correctly?


One way to solve the problem is to use the trial-and-error approach. We may assume the student answered all 25 questions correctly, or 24, or 23, etc. Then check to see if the corresponding points match 80. The following table lists the partial solution:

No. of Correct Answers Points
  25   25 4 - 0 1 = 100
  24   24 4 - 1 1 = 95
  23   23 4 - 2 1 = 90
  :   :
  21   21 4 - 4 1 = 80

What if you started from the other end, that is, 0, 1, 2, etc? How many trials does it takes to get the correct answer? Can you see the shortcoming of this trial-and-error approach?


As shown, this approach works, but it is generally very time consuming, especially when the problem gets more complex. Now, a natural question is "Is there a better approach?" The answer is algebra, of course.

Let's assume that the number of correct answers is represented by a variable named "x". The following is a list of results we can obtain based on this assumption.

        Number of correct answers
            = x

        Number of wrong answers
            = 25 - x

        Total points gained
            = (Number of correct answers) 4
            = 4x

        Total points lost
            = (Number of wrong answers) 1
            = (25 - x)

        Net points
            = 4x - (25 - x)
            = 5x - 25

Since the student obtained 80 points, we have:

        5x - 25 = 80

As shown, we obtained a linear equation with one variable. How do we solve this equation to find the variable x (number of correct answers)? To solve it means that we want to go from the above form to the following expression:

        x = (?)

You might prefer the trial-and-error approach because it is easier to understand, easier than learning a new tool, and it is easier to rely upon your existing arithmetic. You may be right in this case. However, just like walking to your neighbor´s house, your legs may be better than your car. It may not be worth the trouble (getting into the car, getting the car started, getting out of the car) to drive for a few yards. However, can you still claim that your legs are better than the car when you try to get from San Francisco to New York? It is certainly worth spending the time to learn the new tool (how to drive a car) and use it. Similarly, algebra is a much better tool when dealing with more complicated problems.