# Adding and Subtracting Like Terms

There is a very simple property for adding and subtracting algebraic expressions. To be able to add or subtract expressions, we must have like terms. Like terms are terms that contain the same variable or group of variables raised to the same exponent, regardless of their numerical coefficient. For example:

3x and 6x are like terms. They both contain x.

6c2 and 19c2 are like terms. They both contain c2.

2xy3 and 101xy3 are like terms. They both contain xy3.

km2x5 and 17km2x5 are like terms. They both contain km2x5.

Notice that to determine like terms, you must consider the variables in each term as a group. Like terms are those with exactly the same variables raised to the same exponent. If two terms have the same variables, but to different powers, they are not like terms and cannot be combined. For example:

x4 and 3x2 are not like terms since one contains x4 and the other contains x2 as variables.

5vk3 and vk2 are not like terms since one contains vk3 and the other contains vk2 as variables

 Adding and Subtracting Like Terms To combine like terms, do the following: Determine which terms contain the same variable or groups of variables raised to the same exponent. Add or subtract the numerical coefficients. Attach the common variables and exponents.

For example, 3x + 6x can be simplified to (3 + 6) x = 9x

### Example

If possible, simplify each of the following expressions.

1. 10a + 10b –3a
2. 5b2 + 8b3
3. 3x2y2z – 5xyz + x2y2z

1. 10a + 10b – 3a = 7a + 0b
2. 5b2 + 8b3 = 5b 2 + 8b3
3. 3x2y2z – 5xyz + x2y2z = 4x2 y2z – 5xyz

(1) 10a + 10b – 3a = 7a + 10b

To find the solution:

 Determine which terms contain the same variable or groups of variables raised to the same exponent: If we look at this equation we see that there are two terms which contain the variable a. 10a + 10b – 3a Add or subtract the numerical coefficients: We use the distributive property to rewrite the equation. Then perform the subtraction indicated, subtracting 3 from 10. (10 – 3) a + 10b Attach the common variables and exponents: We then display the final result from the subtraction. 7a + 10b

(2) 5b2 + 8b3 = 5b2 + 8b3

 Determine which terms contain the same variable or groups of variables raised to the same exponent: While both terms have b's in them, they are raised to different powers, b2 and b3. This means we cannot combine these two terms. 5b2 + 8b3

(3) 3x2y2z – 5xyz + x2y2z = 4x2y2z – 5xyz

 Determine which terms contain the same variable or groups of variables raised to the same exponent: If we look at this equation we see that there are two terms which contain the variable x2y2z. 3x2y2z – 5xyz + x2y2z Add or subtract the numerical coefficients: We group these terms together and perform the addition indicated, adding 3 and 1 together. (3 + 1)(x2y2z) – 5xyz Attach the common variables and exponents: Here we have the final result. 4x2y2z – 5xyz

Combining like terms is crucial in solving equations. This is a procedure you will use often with algebraic expressions.