Simplifying expressions
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How to simplify algebraic expressions.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to simplify algebraic expressions.   
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Algematics software. Click here for instructions.

Theory:

Algebraic expressions contain alphabetic symbols as well as numbers. When an algebraic expression is simplified, an equivalent expression is found that is simpler than the original. This usually means that the simplified expression is smaller than the original.

There is no standard procedure for simplifying all algebraic expressions because there are so many different kinds of expressions, but they can be grouped into three types:

(a) those that can be simplified immediately without any preparation.

(b) those that require preparation before being simplified.

(c) those that cannot be simplified at all.

 

NOTE: This article will discuss three examples of algebraic expressions that can be simplified immediately.  To simplify more difficult expressions, see the article: Simplify expressions (Advanced).

Example 1:

2x + 3y - 2 + 3x + 6y + 7

This expression can be simplified by identifying like terms and then grouping and combining like terms, like this:

·         +2x and +3x are like terms, and can be combined to give +5x,

·         +3y and +6y combine to give +9y, and

·         -2 and +7 combine to give +5.

So after simplifying, this expression becomes:

        5x + 9y + 5

Example 2:

3b - (4b - 6b + 2) + b

The brackets in this expression can be removed first, then the expression may be simplified like example 1 above, OR the like terms: +4b and -6b inside the brackets can be combined to give -2b before removing the brackets. This is what Algematics will do if you use the simplify command on this expression. (The next article will show you how to make Algematics remove the brackets first.)

·         Simplifying the bracketed expression we have:

3b - (-2b + 2) + b

There is a minus sign before the left bracket, so the sign of each term inside the brackets will change when the brackets are removed:

·         Removing the brackets:

NOTE: This step is not displayed by Algematics unless you use the expand command as explained in the method section below.

 3b + 2b - 2 + b

·         Grouping and combining the like terms:  +3b, +2b and +b combine to give 6b:

 6b  -  2

Example 3:

Expressions that are products and quotients of simple factors that include powers with the same base can be simplified immediately by adding and subtracting the indices of the powers.

This expression simplifies to give:  4p²q³

 

Download the free support file... We have created an Algematics document containing the completed example from this topic. It also includes practice exercises to improve your skills.

File name:  'Simplifying expressions.alg'   File size: 5kb
Click here
to download the file.

If you choose 'Open this file from its current location', then Algematics should open the document immediately. If not, try the other option: 'Save this file to disk', then run Algematics and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Algematics installed on your computer, click here for instructions.

Method:

IMPORTANT: This topic assumes that you know how to enter mathematical formulas into Algematics. Find out how by completing the three simple tutorials in the 'Getting Started' section of the Algematics program 'Help'.

We will use Algematics to simplify each of the example expressions above. Then you will be able to simplify similar expressions yourself.

Step 1  Enter the data

Click  and type your expression into the maths box in the data entry dialog box. If the ‘EMPTY’ message is not displayed between the blue buttons, click the  button until the message: ‘INSERT’ appears.
If necessary, use the ‘ * ’ symbol for multiply, and the ‘ / ’ symbol for divide.

       Maths...

   2x + 3y - 2 + 3x + 6y + 7

 

 

and then click

Step 2  Use the Simplify command

Click  (simplify) repeatedly until the expression is completely simplified. 

If the expression is not changed by the simplify command, then either it is completely simplified, or you need to use another approach. (See the companion topic: Simplify expressions (Advanced) )

NOTE: If you simplify two many times, you will duplicate the last step. You can delete a duplicate step by selecting it, then using Ctrl+Delete on the keyboard.

To display a step where brackets are removed, as for example 2 above, click  (expand).

 

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