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 color 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 a six 𝑥 plus four 𝑦 plus
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’re also plus five 𝑦𝑥. And because multiplication is
commutative, that means 𝑦𝑥 is the same as 𝑥𝑦. So I am 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 startup 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 the three different flavors 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 an algebraic expression down
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 twelve of something. So we’ve got twelve lots of 𝑥 plus
Now we can use the distributive property of multiplication to say, that means
twelve times 𝑥 plus twelve times two. And we can write that algebraically as twelve 𝑥, short for
twelve times 𝑥. And twelve times two is twenty-four, so
twelve 𝑥 plus twenty-four. 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 ten plus seven 𝑥 plus fourteen, which means we can now gather the like terms. Five 𝑥 and seven 𝑥
are like terms. And then the numbers on their own, ten plus fourteen, which gives us twelve 𝑥 plus twenty-four. 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 eleven 𝑥 plus
twelve 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 eleven 𝑥 plus
twelve to five, as if we add twelve to the result of
five plus eleven 𝑥. 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 twelve gives us
seventeen. So our answer is seventeen plus eleven 𝑥, or as some people
prefer to write it with the letter term first eleven 𝑥 plus seventeen.
Now one last thing before we go. We’ve dealt with positive and negative
quantitative terms like three 𝑥, minus five 𝑦, minus twelve 𝑦
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 give 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 summarize what we’ve learnt then. We can simplify
algebraic expressions by combining or collecting like terms. For example, three 𝑥 minus twelve 𝑦 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 ten 𝑥, while negative twelve 𝑦 plus two 𝑦 gives us negative ten 𝑦.
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
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’ve 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 twelve, 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.