Video Transcript
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.
