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.

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

Answers

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

Detailed Answers

(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.


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