In this video, we’re gonna be looking at how to simplify algebraic expressions by combining like terms. We’ll be looking at expressions that contain numbers and letters and combinations of letters and thinking about the effect of parentheses on some of the expressions.
We can simplify algebraic expressions by combining or collecting like terms. For example, all the numbers on their own, all the 𝑥s, all the 𝑦s, and so on.
So simplify two plus three 𝑥 plus two 𝑦 minus 𝑥 plus five 𝑦 plus four 𝑥 minus three 𝑦 plus seven. So thinking about the purely numeric terms first, we’ve got a two and we’ve got a plus seven. And then looking at the 𝑥 terms, we’ve got a plus three 𝑥, take away 𝑥, add four 𝑥. And that leaves us with our 𝑦 terms, plus two 𝑦, plus five 𝑦, take away three 𝑦. Some people find it useful to use either a colour code system or a system of different symbols to keep track of which like terms they’ve encountered and they’ve used so far and which ones they haven’t.
So dealing with the purely numerical terms first, two plus seven is equal to nine. Now we’ve got our 𝑥 terms. We’ve got ne- three 𝑥 take away 𝑥 is two 𝑥, add four 𝑥 is six 𝑥, so that’s plus six 𝑥. Then the 𝑦 terms, we’ve got two 𝑦 plus five 𝑦 is seven 𝑦, take away three 𝑦 is plus four 𝑦. And most people tend to put the letter terms first and the purely numerical terms at the end of that expression. So we’ll say it’s six 𝑥 plus four 𝑦 plus nine.
But sometimes, terms involve more complex terms such as 𝑥𝑦 or 𝑦 cubed. And we need to be able to identify like terms in these expressions too.
For example, simplify three 𝑥 plus four 𝑥 squared minus two 𝑥𝑦 plus seven 𝑦 squared minus two 𝑥 plus five 𝑦𝑥 minus 𝑦 cubed minus two 𝑥 squared.
Now in this expression, there’re two terms, three 𝑥 and take away two 𝑥, which are purely numbers times 𝑥. So let’s put those first. And then if we look for the 𝑥 squared terms, we’ve got plus four 𝑥 squared and minus two 𝑥 squared. Now let’s look for 𝑥𝑦 terms. Well, there’s minus two 𝑥𝑦. But there’s also plus five 𝑦𝑥. And because multiplication is commutative, that means 𝑦𝑥 is the same as 𝑥𝑦. So I’m just gonna write five 𝑦𝑥 as five 𝑥𝑦.
Next, we’ve got 𝑦 squared terms. Well, there’s only one of those, seven 𝑦 squared. And we’ve dealt with the others, apart from the 𝑦 cubed. So we’ve just got the minus 𝑦 cubed. So like terms are terms which are just simple multiples of the same letter, or the same letter to the same power or exponent, or the same combination of letters multiplied together, so 𝑥𝑦, 𝑦𝑥. And three 𝑥 take away two 𝑥 is just one 𝑥, or just 𝑥. Four 𝑥 squareds take away two 𝑥 squareds just leaves us with two 𝑥 squareds.
Now if we start off with negative two 𝑥𝑦s and then we add five more 𝑥𝑦s to that, we’ll end up with positive three 𝑥𝑦s. Then we’ve just got positive seven 𝑦 squared and minus 𝑦 cubed. Now none of those are like terms, so that is our answer. So it’s almost as if 𝑥 is treated as a different letter to 𝑥 squared. And that’s treated as a different letter to 𝑥𝑦 and 𝑦 squared and 𝑦 cubed, and so on.
Let’s look at this visually and see why we’ve defined the term “like terms” in that particular way. Imagine we’ve got some one-by-𝑥 rectangles and some one-by-𝑦 rectangles. So a one-by-𝑥 rectangle, one times 𝑥, has an area of 𝑥. So we’ll call these ones 𝑥. And one-by-𝑦, one times 𝑦, has an area of 𝑦. So these are 𝑦 rectangles.
So I’ve got three of the 𝑥 rectangles and two of the 𝑦 rectangles. So we can write that as three 𝑥 plus two 𝑦. Now 𝑥 rectangles are a different size to 𝑦 rectangles. So we can’t really combine them. It’s just a statement of the fact that we’ve got three of the 𝑥 style and two of the 𝑦 style.
Now let’s add in an 𝑥-by-𝑦 rectangle. So that’s- area of that would be 𝑥 times 𝑦, so we’ll call that 𝑥𝑦. Now we’ve got three of the 𝑥 rectangles, two of the 𝑦 rectangles, and one of the 𝑥𝑦 rectangles. So we have to write this like this. We can’t simplify that down any further. It’s just a statement of our three different flavours of rectangle, 𝑥s, 𝑦s, and 𝑥𝑦s. And that tells us how many of each we’ve got.
Now we’ve added in to the mix three 𝑦-by-𝑦 rectangles, 𝑦 squared rectangles, and one 𝑥-by-𝑥 rectangle, so an 𝑥 squared rectangle. So now we’ve got three 𝑥 plus two 𝑦 plus 𝑥𝑦 plus an 𝑥 squared and three of the 𝑦 squared rectangles. So hopefully, we can see that an 𝑥 is quite different to an 𝑥 squared. And a 𝑦 is quite different to a 𝑦 squared. And an 𝑥𝑦 is something completely different again. That’s why we have to express them separately in our algebraic expression down here.
Now let’s take a look at some expressions which involve parentheses. For example, simplify five times 𝑥 plus two plus seven times 𝑥 plus two. And that really means we’ve got five lots of 𝑥 plus two. Imagine an 𝑥 plus two stock size rectangle. And we’ve got seven lots of 𝑥 plus two. So we’ve got five of these 𝑥 plus two rectangles, and we’ve got seven more of these 𝑥 plus two rectangles. So 𝑥 plus two in parentheses are like terms. And five of something plus seven more of something make 12 of something. So we’ve got 12 lots of 𝑥 plus two.
Now we can use the distributive property of multiplication to say that means 12 times 𝑥 plus 12 times two. And we can write that algebraically as 12𝑥, short for 12 times 𝑥. And 12 times two is 24, so 12𝑥 plus 24.
Now we could’ve applied the distributive property earlier in that question. So that would’ve meant five times 𝑥 and five times two and seven times 𝑥 and seven times two. So that will be five 𝑥 plus 10 plus seven 𝑥 plus 14, which means we can now gather the like terms. Five 𝑥 and seven 𝑥 are like terms. And then the numbers on their own, 10 plus 14, which gives us 12𝑥 plus 24. So it gives us the same answer.
But by spotting the fact that we had this like term here, 𝑥 plus two. It meant that we only had to do one lot of distributing here, rather than two lots of distributing here. So this method here involved a bit more working out. We saved ourselves a bit of work by spotting this like term here, in the first place, in the parentheses.
Now some expressions contain parentheses, which when we carefully examine them can actually be removed. For example, simplify five plus parentheses 11𝑥 plus 12, close parentheses. Now because this expression uses addition in two places, and addition is associative, we will get the same result if we add the result of 11𝑥 plus 12 to five as if we add 12 to the result of five plus 11𝑥. So the parentheses are telling us to do the calculation one way. But it doesn’t make any difference if we do it the other way. They’re just redundant. So we can erase them.
And now we’ve got two like numeric terms. And five and 12 gives us 17. So our answer is 17 plus 11𝑥, or as some people prefer to write it with the letter term first 11𝑥 plus 17.
Now one last thing before we go. We dealt with positive and negative quantitative terms like three 𝑥, minus five 𝑦, minus 12𝑦 squared, and so on. But we can also deal with fractional quantities of terms in just the same way. For example, a half 𝑥 plus another half of an 𝑥, well two halves of the same thing gives us a whole one of the same thing. So a half 𝑥 plus a half 𝑥 is one 𝑥, or just 𝑥, as we would now more normally write it.
So for example, if we have to simplify minus a third 𝑥 plus two-thirds 𝑥 plus a quarter plus two-thirds 𝑥 plus a half, we can identify our like terms. So we’ll bring all the 𝑥 terms together. And we’ll bring the numeric terms together as well. So we’ve got negative a third 𝑥 plus two-thirds 𝑥 plus another two-thirds 𝑥 plus a quarter plus a half.
Let’s think about those 𝑥 terms then. We’re starting off on negative one-third of an 𝑥. And we’re adding two-thirds of an 𝑥. So one-third, two-thirds of an 𝑥 brings us to here. And then we’re adding another two-thirds of an 𝑥. So we’re gonna go one, two-thirds of an 𝑥 is gonna bring us to here, at one. So that’s one 𝑥, or just 𝑥. And then we’ve got one-quarter plus one-half. Well, a half is two-quarters. So we’ve got one-quarter plus two-quarters, which is three-quarters. So the same rules apply. Even if you’ve got fractions, you can still collect or combine like terms in algebraic expressions to simplify them.
So finally, just to summarise what we learned then, we can simplify algebraic expressions by combining or collecting like terms. For example, three 𝑥 minus 12𝑦 plus seven 𝑥 plus two 𝑦. We can treat the 𝑥 terms as like terms and the 𝑦 terms as like terms. And three 𝑥 plus seven 𝑥 gives us 10𝑥, while negative 12𝑦 plus two 𝑦 gives us negative 10𝑦.
But remember, terms with different exponents or powers are not like terms. For example, five 𝑥 plus seven 𝑥 squared minus two 𝑥 minus three 𝑥 squared has five 𝑥 and minus two 𝑥 as like terms and seven 𝑥 squared minus three 𝑥 squared as like terms. So when I combine them, five 𝑥 minus two 𝑥 gives us three 𝑥. And seven 𝑥 squared minus three 𝑥 squared gives us four 𝑥 squared. It’s as if 𝑥 squared is a different letter to 𝑥. Just think about those different size rectangles that we showed you earlier when we had the visual representation of this. We’re gathering together all of the 𝑥 size rectangles and we’re gathering together all of the 𝑥 squared size rectangles.
And finally, sometimes there are whole parentheses containing like terms. For example, three times 𝑥 minus two plus five times 𝑥 minus two minus two times 𝑥 minus two. These terms are all multiples of 𝑥 minus two. So we’ve got three of them and another five of them. But we’re taking away two of them, which leaves us with six of these 𝑥 minus twos. This means that I can apply the distributive property just once at the end of the question to get six 𝑥 minus 12. Rather than three times earlier in the question and then having to gather lots and lots of terms together. So sometimes spotting this fact can save us a bit of work.