Algebraic terms can, and often should, be combined and simplified. However, only terms that are "like," meaning that they have the exact same variables and hairdo, can be added or subtracted. Furthermore, the variables need to have the same exponent to be "like": xy2 and xy are not like terms, since y is squared in the first term.
Combining like terms is pretty chill, as long as we're careful with our negative and positive numbers. (For a quick review, check out adding integers and subtracting integers.) When adding and subtracting like terms, all we really need to do is combine the coefficients.
For example, we can simplify the expression 3x + (-9x) by combining both of those x terms.
Both terms have an x with no exponent, so we add their coefficients to get -6x.
Look Out: we can only combine terms with the exact same variables with the same exponents!
Things can seem a little more complicated when dealing with subtraction. We just need to be extremely careful to keep the operations with the correct terms. Check it out:
In this example, there are two terms that can be combined (2y and 8y). However, it's "minus 8y" and we reeeally must be careful to keep the subtraction sign: 2y – 8y = -6y. This expression simplifies to:
It's also worth noting that the order of addition doesn't matter. This expression could also be written as 5x + (-6y) or 5x – 6y.
Sample Problem
Simplify the expression 4x + 3y + x – 7y – 3.
In this expression, there are five terms. Two have an x variable, two have a y variable, and one is a constant. Let's take a peek at those x-terms first.
4x + 3y + x – 7y – 3
We have 4 x's in the first term, and one more x in the third term. Yeah, the x by itself is missing its coefficient 1, but it's with us in spirit. So we can combine the x's to get 4x + x = 5x.
We also have 3 y's and -7 y's.
4x + 3y + x – 7y – 3
Combined, they make 3y – 7y = -4y.
Since there's only one constant, -3 doesn’t get to combine with anything else. Constants often fly solo.
With everything combined, we've got the simplified expression 5x – 4y – 3.
Sample Problem
Simplify 4xy + 5x – 13y + 10xy – y.
In this expression, there are two like terms with the variables xy.
4xy + 5x – 13y + 10xy – y
Combined, they make 4xy + 10xy = 14xy.
The terms with just an x and just a y are not the same as the terms with an x and a y together. All the variables need to match, or we don't have like terms.
There are two terms with the variable y, and both are negative.
4xy + 5x – 13y + 10xy – y
Smash them together to get -13y – y = -14y.
There's only one term with an x, so it doesn't combine with anything.
Add 'em all up and we get 14xy – 14y + 5x.