The phrases “combine like terms” or “collect like terms” are used a lot in algebra, and for good reason. The process they describe is used a lot in solving algebra problems. Two approaches, one intuitive and the other algebraic, can help in understanding why some terms are “like” terms, and others are not.
Quantities With Units
Suppose you are sitting in front of a table that holds three piles of fruit:
– five apples
– three oranges
– four apples
If someone asks you “What do you see on the table?”, how would you answer the question?
Chances are you answered “nine apples and three oranges”. Why did you combine the two piles of apples with one another, but not with the oranges? How did you know that you could do that?
The quantities of apples may be combined because addition or subtraction only work with quantities that have the same units, such as apples in the example above. If you were asked to add three apples and two oranges, the only answer you could give is “three apples and two oranges”, since neither addition nor subtraction make sense with quantities that have differing units. The concepts of addition and subtraction are based on combining or separating quantities with the same units.
However, if you are allowed to change the units, you might be able to convert one or more quantities from their current units into a common unit. For example, if you called both the apples and the oranges “pieces of fruit”, the problem would become “add three pieces of fruit and two pieces of fruit”, at which point you could combine the two quantities to arrive at an answer of “five pieces of fruit”. When adding inches and centimeters, you could convert both quantities into inches, or both into centimeters, at which point you would be able to add the quantities.
Quantities with different units cannot be combined using addition or subtraction. Three meters plus two rocks? How could one possibly add a length measurement to a quantity of physical objects? It does not make sense because they are “unlike” quantities. Addition and subtraction only make sense with “like” quantities: quantities that either all have the same units (2 feet and 3 feet), or all have no units (scalar quantities like 2 or 3).
Terms are the quantities in an expression which are being added or subtracted from one another. This definition can be applied to an expression as a whole, as well as within groupings inside an expression. The following examples should help clarify this:
has two terms, the 2 and the 3.
is a single term, and is also a fraction. The entire expression can be considered as one term, but it can also be described as a fraction or division problem with one term in its numerator (which is “grouped” together by the fraction line) and one term in its denominator (which is also considered “grouped”).
is a single term and a fraction once again. In this example, the “grouping” effect of the fraction line on the numerator and denominator is hopefully more clear… if you put parentheses around the entire numerator or denominator, they would not change how this must be evaluated. This expression could be described further by saying that the numerator contains two terms, and the denominator three terms.
is a single term which is a product. Since “factors” are quantities which are being multiplied by one another, we could describe this expression further by saying that the first factor in the product contains two terms, and the second factor contains three terms.
has two terms. The first term in the expression is a product of three factors (two, x, and y), and the second term in the expression is a product of a binomial (a two term polynomial) and a monomial (a one term polynomial).
Generalizing from the above examples:
- within an expression or a grouping symbol (parentheses, fraction lines, radicals, etc.), terms are separated by addition or subtraction symbols
- it is sometimes useful to view an entire expression as a series of terms
- it is sometimes useful to look within a term or a factor, particularly when it contains grouping symbols, to see if it contains more than one term
Like Terms: Intuitively
“Like Terms” are terms that can be combined using addition or subtraction (when the order of operations permits). For terms to be combined using those operations, the terms must have the same units. For example:
Constant terms with no units (scalar quantities) are also “like terms”, and may be combined because they have the same lack of units. For example:
But what about terms that include more than just constants? For example, terms which include variables:
Since, in general, a variable is a placeholder for a value we do not know yet, that means that we do not know the units of the variable’s value either. In the above example, until we are given values (with units) for the variables, we have no idea what the units for “x”, or “y”, or “xy” will be… so we cannot combine them (yet?). If, after being given values for the variables, they turn out to have the same units (or lack of units), then we would be able to combine them – however, if they turn out to have differing units, then we could not combine them.
If, on the other hand, two terms contain exactly the same variables, such as:
then both terms must have the same units when a value is substituted for the variable “b”, since the same quantity with the same units will replace every instance of “b” in the expression. Therefore, terms which contain exactly the same variables will have to have exactly the same units, and will be able to be combined using addition or subtraction. They will be “like” terms.
Like Terms: Algebraically
If two or more terms in a sum or difference share a common factor, then those terms can be re-written as a product of that factor and a sum, then simplified:
The common factor (apples) has been factored out of both terms. We have, in effect, used the distributive property of multiplication over addition backwards. You can verify for yourself that if you carry out the multiplication indicated on the right side of the equal sign above, you will end up with the expression on the left side of the equal sign.
After factoring out the common factor (apples), note that you have two constant terms in the parentheses with no units. Since we know that 2+3 is equal to five, we can use substitution to replace the expression (2+3) with (5)… something that means exactly the same thing as (2+3).
The exact same reasoning can be applied to a situation that involves a variable (“x” in this example) instead of units:
Note that a “common factor” is not enough to create like term though… the terms left behind after factoring all must still have the same units or lack of units. Consider a more complex situation:
Note that the terms left behind after factoring out the “x” (2 and 3y) are not like terms (may not have the same units once a value for “y” is specified), therefore the expression cannot be further simplified. This means that the two terms in the original expression, 2x and 3xy, are not “like terms” even though they share a factor.
All of this leads to a formal definition of “like terms”:
“Like” terms each have the same variables to the same powers
The term “2x” did not have exactly the same variables as the term “3xy”, therefore we do not even have to think about their powers to be able to state that they are not “like terms”.
Both terms have the same variables (x) to the same powers (1), therefore they are “like” terms and may be combined:
The terms have different variables, and therefore are not “like” terms.
The terms have the same variables, but the “x” is raised to the first power in the first term, and to the second power in the second term. Since the powers of the variable are different, they are not “like” terms.
The terms have the same variables, but the first term could be simplified a bit since the “x” is being multiplied by itself:
This makes it easier to see that both terms have the same variables to the same powers, therefore they are “like” terms:
The terms both have “x” in them, but one has a “y” and the other does not, therefore the two terms do not have exactly the same variables and are not “like” terms.
The terms both have “x” and “y” in them, but their powers are not identical. “x” is squared in one term, while “y” is squared in the other, therefore the two terms do not have the same variables to the same powers, and the terms are not “like” terms.
This expression has two terms separated by subtraction. If you could drop the parentheses, you can anticipate two pairs of like terms, the “x” terms and the “y” terms, however the order of operations and the parentheses prevent us from combining them as they are written. The negative sign in front of the second set of parentheses must be distributed before we can drop all parentheses, and it will change the sign of all terms inside the second set of parentheses:
Now like terms may be combined:
Only quantities that have the same units may be added or subtracted. This is a fundamental property of addition and subtraction which makes intuitive sense.
If two terms contain the same variables to the same powers, then the two terms must also have the same units – no matter what units each variable represents.
In an expression that contains multiple “like” terms, the commutative property of addition may be used to arrange all “like” terms adjacent to one another in the expression, then factoring (using the distributive property of multiplication over addition and subtraction backwards) and substitution can be used to “collect”, or “combine”, all like terms into a single term that is equal to their sum or difference.
“Combining like terms” produces an equivalent expression with fewer terms, a”simpler” expression. Therefore combining like terms is a frequently used step in simplifying expressions and equations.
If a formal algebraic approach were always taken when collecting like terms, the work might look something like this:
Commutative property of addition
Distributive property of multiplication
Addition property of equality
Subtraction property of equality
Distributive property of multiplication
Division property of equality
That sure felt like a lot of work compared to the usual way of simplifying an expression. I included the above to help illustrate the algebraic approach to “why” collecting like terms produces an equivalent expression. Once comfortable with the idea and process of collecting like terms, most people do not bother taking each of the above steps individually, let alone justifying each with axioms or theorems. Once you are able to recognize like terms easily, multiple steps are often taken at once to combine “like” quantities:
Combine like terms on each side
Add”2x” to each side, then combine like terms
Subtract “8y” from both sides, then combine like terms
Divide both sides by 3
The expression “x = -y” is a simplified version of the initial expression. We know that it is “as simplified as possible” because it no longer contains like terms or common factors.