Combining Like Terms
Another way we sometimes solve equations is by combining like terms. Combining like terms means putting all the numbers with the same variables together. Think about doing something in your life, like cleaning your room. You probably put the board games together, the video games together, the clothes together, all the pillows and blankets together . . . and so on. In a way, you are combining the like “terms” of your room.
In other words, an equation might come to you looking like this:
2x + 3y + 4x = 9
If you looked at the terms, or variables, of the equation, you would see that there are two x terms, 2x and 4x, and they’re being added together (you know this because the 4 has a + in front of it. If it were a – sign, we would know they would be subtracted. Now, you can combine the two like terms, which would look like this:
2x + 4x = 6x
Now, you have 6x + 3y = 9. This is as much as you can solve this equation, because you are not given values for either of the variables. Thus, your final answer is 6x + 3y = 9.
Here’s one for you to try:
8n – 9m + 3n = 12
First, combine the two n terms: 8n + 3n = 11n
Then, put the combination back into the equation: 11n – 9m = 12
And, here’s one last example—this one’s a tricky one!
9x + 2b + 8y – 3x +2y – 10b = 5
First, let’s put all the like terms together so that we don’t get confused. Remember, when we move a variable, we move the + or – sign that is in front of it as well.
9x – 3x
2b – 10b
8y + 2y
Now, we solve for each of them, like this:
9x – 3x = 6x
2b – 10b = – 8b
8y + 2y = 10y
Now, we take the “answers” we just got, and put them back together, like this:
6x – 8b + 10y = 5
We know that the 8b is negative, because we see the big minus sign in front of it, but how do we know that the other two are positive? Well, when combining like terms, if there is no negative sign in front of the variable, we can assume that it’s positive; therefore, you’ll stick a plus sign (+) in front of it.
Here are a couple examples for you to try. Combine like terms, and then pick the answer that best matches yours.
8x – 9y + 4x + 10y
First, rewrite the equation so that like terms are together, like this: 8x + 4x – 9y + 10y
Then, combine the x terms by adding 8x + 4x and combine the y terms by adding – 9y + 10y
Last, put the two terms back together, to get 12x + 1y or just 12x + y
5c + 7d – 2c +1d
First, rewrite the equation so that like terms are together, like this: 5c – 2c + 7d + 1d
Then, combine the c terms by subtracting 5c – 2c and combine the d terms by adding 7d + 1d
Last, put the two terms back together, to get 3c + 8d
Combining Like Terms with Exponents
Now that you have gone through basic combining like terms, you can move on to harder equations. So far, the equations we’ve looked at have used either one or two variables, without exponents. As you move on into the study of algebra, however, you will be asked to combine like terms with exponents, which you need to do very carefully. We’ll walk you through this process so that you don’t get confused.
If you saw this equation: 4 + x – 3 + 2x = 10, you would combine the 4 and –3, and the x and 2x, to get 2x + 1 = 10. However, if the equation looked like this: 4 + x2 – 3 + 2x = 10, what would you do?
The right thing to do would be to combine only the like terms, which are the 4 and – 3, so your equation would look like this: x2 + 2x + 1 = 10. Your first instinct might be to go ahead and add the x terms together, BUT this would be a bad thing to do! You cannot combine the x2 and 2x, because the first term has an exponent (2) and the second one does not have an exponent; therefore, you cannot add them together! When combining like terms, you must make sure than all the variables, and exponents, are the same before you add them together.
Let’s practice with a few more of these, and then you can try it on your own.
Combine like terms from the following equation:
5x + 7x2 – 6 + 10 – 4x + x3 = 12
It’s always good practice, when combining like terms and solving equations, to list the terms from the variables with the highest exponents to the variables with the lowest exponents, followed by a number without a variable (whole numbers, fractions, and so on). We’re going to use this practice in re-writing all of our equations.
In order to combine like terms, we identify the term(s) with the highest exponents. In this case, we have one term, x3, with an exponent of three. Since this is the highest, we’ll write it first. Then, identify the term with the next highest exponent, which is 7x2. There are no more variables with an exponent of 2, so we’ll write this one after our first term. Next, we look for x variables, and find that we have 5x – 4x, so we can combine these terms (by subtracting) and end up with simply + x. So far, our terms read: x3 + 7x2 + x. Now, we have to check for any whole numbers we can combine. We see that we have – 6 + 10, which we can combine to give us a + 4. We write that after our other terms, for a final equation of x3 + 7x2 + x + 4 = 12.
Notice that this equation did not combine a lot—only 4 terms were combined (two x terms and two whole numbers). This is perfectly fine, and will often happen in algebra. Don’t worry, it doesn’t need to look super-simplified in order to be considered correct. Sometimes you can only combine one or two things before you’re done with an equation.
Let’s look at another one. Combine like terms in the following equation:
8x2 + 9x3 – 3 + x2 – 4x2 + 3x + 5 – 9x – 10 = 3
This may look intimidating, but if you take it apart step by step it’s not bad. Remember, start by looking for the variable with the highest exponent, and then work your way down from there. We look for the highest exponent, which in this problem is 3, and see that there is only one term with an exponent of 3. Therefore, we write 9x3 as our first term in the combined equation. Then we look at the next highest exponent, 2, and see that there are three terms with an exponent of two; they are: 8x2, x2, and – 4x2. The first two terms are positive, so they get added together resulting in 9x2, and the last one is negative, so we subtract the term, leaving us with 5x2. Our next term is simply the x term, which we have two of; they are: 3x and – 9x. We combine these through subtraction, and get –6x. Lastly, we look at our whole numbers; there are 2 of them, and they are: 5 and –10, which we subtract to get –5. Now, we list all of these terms together to get 9x3 + 5x2 – 6x – 5 = 3.
Combining Like Terms with Exponents Quiz
7 + 8x – 9x2 + 2 + 5x3 – 3x2 + x + 16 = 3
First, find the highest exponent, which is 3. There is only one variable with an exponent of 3, so we write that one first (5x3). Then, we find all of the variables with exponents of two; there are two of them: – 9x2 – 3x2, which equals –12x2. Next, look for the x terms; there are two of them: 8x and x. We combine these by adding these, giving us 9x. Lastly, we gather all of the whole numbers without variables together; they are: 7 + 2 + 16 and we add them together to get 25. Therefore, our final combined answer is 5x3 – 12x2 + 9x + 25 = 3
Now, we’re going to give you one more problem. Be careful, it’s a tricky one. Hint: remember that you can only combine things that have variables that look exactly the same; in other words, if you had a2b + ab2, you cannot add those together!
x2 – 5xy2 + 2x2y + 3xy + 4x2y + 7xy2 – xy – y2
For these types of equations, the best practice is to list the x variables with exponents first. After looking up and seeing that we have one x-variable with an exponent (x2), we know that will come first. Then, the next term you should deal with is the xy variable, when x has an exponent. We look up and see that we have two x2y terms: 2x2y + 4x2y, which gives us 6x2y. Next, the term you should deal with is the xy variable, when y has an exponent. We look and see that we have two xy2 terms: – 5xy2 + 7xy2 which gives us 2xy2. Next, we take care of the xy variable without any exponents; there are two of these. 3xy – xy which gives us 2xy. Lastly, we deal with the y term, of which there is only one, which is – y2. Since there is only one y2 term, we can’t combine it with anything, so we simply leave it alone.
Then, we put all of our terms together, and get the following:
x2 + 6x2y + 2xy2 + 2xy – y2