Linear Equations Lesson

Linear equations are equations involving only one variable, like x or y, and they do not involve anything complicated like powers, square roots, or anything like that.

Some people think that since linear equations are the simplest equations that students encounter, they are the easiest to learn. But nothing could be further from the truth. Linear equations are hard to learn and their reputation is well deserved. The reason for this is that linear equations are the first opportunity to practice operating with equations and develop equation solving skills. So if you are frustrated with linear equations, you are most likely not stupid and in a good company.

Linear Equations in Word Problems

Bob has three sacks of apples and three more apples in his pocket. Each sack contains the same number of apples. Altogether, Bob has 33 apples. How many apples are in each sack?

This easy word problem captures the essense of all linear equations. Whenever you are stuck with some linear equation, just remember Bob and his apples. How do you solve it? Let us write the solution in words, so that you would not be bothered with mathematical forms. All you need to know is your arithmetics.

First of all, have Bob eat the three apples from his pocket. How many apples would be left? If you still remember arithmetics, since he had 33 apples and ate 3, he would have 33-3 apples left, that is, 30 apples would be left in sacks. I hope that you now feel that you are making progress.

What you know is that you have three equal sacks, which altogether have 30 apples. How many apples are in each sack? As should be apparent, you have to split 30 into three equal parts, that is divide 10 by 3. The result is 10. This means that each sack has 10 apples. Here's your answer!

Now let's go back to our friend Bob and his apples. Let's try to rewrite the problem in form of an equation.

As you can see, the first thing you do in problems like that is to write an equation. This equation simply restates the problem, but in mathematical language instead of English. Because in the old times ink was expensive, mathematicians did not like to write long names like "apples", so they use letter x or some other letters instead of "apples" and such.

To solve equations like this, you have to get rid of whatever surrounds x. What you want is x to be left by itself on one side of the equation, and the rest to be gone.

Solving Linear Equations

To solve the equation above, first subtract 3 from both sides. This operation is the mathematical equivalent of Bob eating the three apples that were in his pockets. What will remain from the equation is:

My hope is that by now you know what to do next: to get the value of x (which means find out the quantity of apples in each sack), you have to divide the left and the right part of the equation by 3. This mathematical operation is the equivalent of the word solution that I provided one page above.

This is really about it. A very important point is this: No matter what kind of equation you're dealing with -- linear or otherwise -- whatever you do to the one side, you MUST do the exact same thing to the other side! Equations are like little children: You have to be totally, totally fair! You cannot subtract 5 from one side of the equation and forget to do it with another side.

The illustration below, taken from Purplemath , illustrates how you do subtraction.

Note that sometimes instead of subtraction, you need to do an addition. Example:

I strongly recomment that you Practice Solving Linear Equations here. Doing about 10 equations should be enough.

Graphing Linear Functions

Graphing linear Functions is very easy. Click here to find out how to do it .