Explanation of Combining Like Terms in an Expression - Succeed in Understanding Algebra. Also refer to simplify, operation, number, variable, sub-expression, mathematics, math, maths, Ron Kurtus, School for Champions. Copyright © Restrictions
Combining Like Terms in an Expression
by Ron Kurtus (19 January 2008)
Some expressions can be complex and have a mixture of terms. You would like to simplify them as much as possible, in order to perform other operations, solve an equation or such. One method to simplify an expression is to look for terms that can be added together. This includes numbers, variables and sub-expressions.
Questions you may have include:
- How do you combine numeric like terms?
- How do you combine variable like terms?
- How do you combine like sub-expressions?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Combining numeric like terms
In an expression, the first thing to look for are numbers or numeric terms that stand alone and then combine them in order to simplify the expression.
Consider the expression 2 + 3x − 4xy + 7 − y − 5.
You can see the numbers 2, 7 and −5 can be combined, resulting in a simpler expression 3x − 4xy − y + 4.
Note that typically the numeric terms are listed last in an expression. Also not that y − 5 can be thought of as y + (− 5).
Another example of combining numbers is (x2 + 2x + 3) − 6 + (y + 8).
Although the parentheses group sub-expressions for some reason, removing them does no harm. The numbers can be combined, resulting in the expression x2 + 2x + y + 5.
Combining variable like terms
When simplifying an expression, the next thing to look for are like variable terms. That means any multiplies of a variable or products of variables.
For example, x, 5x and −2x are like terms of x.
Also, 2x2y, −x2y and 5x2y are like terms of x2y.
To get a little tricky, 3yx2 is also a like term of x2y, since you can rearrange terms in a multiplication, such that 3yx2 = 3x2y.
5x and 5y are not like terms and do not combine. Likewise, 2x2y and 2x3y are not like terms.
Example
Consider the expression 3x − 2y − 4xy + x2 + 4y − x + 2xy − y2.
The best way to combine like terms is to first group them together.
(3x − x) +(−2y + 4y) + (−4xy + 2xy) + x2− y2
Note that you can consider an expression like 4y − x to be 4y + (−x). That makes things easier.
Now combine the like terms.
2x + 2y −2xy + x2− y2
Rearrange the terms in a better format.
x2 + 2x −2xy + 2y − y2
Combining like sub-expressions
Sometimes an expression contains multiples of like sub-expressions that can be added together.
Consider 3(x + 7) + 7xy + 4(x2 + 7) + 2xy − 2(x2 + 7) + (x + 7).
Like terms are multiples of (x + 7), (x2 + 7) and xy.
Group the multiples of like terms or sub-expressions:
(3 + 1)(x + 7) + (7 + 2)xy + (4 − 2)(x2 + 7)
4(x + 7) + 9xy + 2(x2 + 7)
We can simplify this even more by completing the multiplication and combine like terms again.
4x + 28 + 9xy + 2x2 + 14
4x + 42 + 9xy + 2x2
Rearrange to put in a better form.
2x2 + 4x + 9xy + 42
Summary
Complex expressions have a mixture of terms that you would like to simplify. One method to simplify an expression is to look for terms that can be added together. This includes numbers, variables and sub-expressions.
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Resources
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Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.
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