### Video Transcript

In this video, we’re gonna look at
how to evaluate algebraic expressions, given specific values of their variables. Sometimes you have to think quite
carefully about the order of operations when looking at algebraic expressions. And we’ll see a couple of examples
where you need to really think about what you’re doing. Let’s look at this example
then.

Evaluate seven 𝑥 plus five when 𝑥
is equal to 10.

So we’re told that 𝑥 is equal to
10 and wherever we see 𝑥 in our expression, we have to replace that with 10. But we have to think quite
carefully. We don’t just literally write 10
because that would then be 710 plus five. Well, that seven 𝑥 means seven
times 𝑥. So we have to think about what the
expression actually means before we substitute those numbers in. So seven 𝑥 plus five means seven
times 𝑥 plus five. So when 𝑥 is 10, that means seven
times 10 plus five. Now, if you think about your order
of operations, you do multiplication and division before you do addition and
subtraction. So we’re going to evaluate this
first. Now, seven times 10 is 70. So that’s 70 plus five, which gives
us an answer of 75.

Now, evaluate negative eight times
𝑚 minus nine when 𝑚 is equal to five.

So we need to substitute 𝑚 is
equal to five into this expression. And that gives us negative eight
times five minus nine. Now, thinking about our order of
operations, we’ve got negative, we’ve got times, and then we’ve got some parentheses
or brackets. So this is what we have to evaluate
first. And five take away nine is negative
four. So that becomes negative eight. Let’s just put some parentheses
around there to make it clear that that’s a negative eight. Negative eight times negative four,
or eight times four is 32. And a negative times a negative
makes a positive. So our answer is 32.

Now, in this question, we’ve got to
evaluate the expression five 𝑎 minus two 𝑏 when 𝑎 has the value nine and 𝑏 has
the value negative three.

So now, we’ve got two variables in
our expression, and one of them has a negative value. So when I have negative values, I
tend to put them in parentheses like this to make it clear that they’re a negative
number. Now, the five right up against the
𝑎 in that expression means five times 𝑎. And 𝑎 has the value nine, so
that’s five times nine. Then we’re gonna be subtracting two
times 𝑏, so that’s two times negative three. Now, we don’t really have to worry
about evaluating these-these brackets or parentheses here, we already know that it’s
got a value of negative three.

So we’ve got two different types of
operation, multiplying and subtracting. And we have to do the multiplying
before we do the subtracting here. So we’re gonna do the five times
the nine, and we’re gonna do the two times the negative three. So five times nine is 45. And then we’re subtracting two
times negative three, so that’s negative six. Now, if we’re subtracting negative
six, that’s the same as adding six. And 45 add six is 51.

Now, let’s evaluate seven 𝑥
squared minus four 𝑦 cubed when 𝑥 is equal to negative two and 𝑦 is equal to
negative three.

So first, remember, seven 𝑥
squared means seven times 𝑥 squared, and four 𝑦 cubed means four times 𝑦
cubed. Now, we can substitute the values
in for 𝑥 and 𝑦, that we were given in the question. So we get seven times negative two
all squared minus four times negative three all cubed. So let’s evaluate these parentheses
first. Negative two all squared means
negative two times negative two. And the negative times a negative
makes a positive, and two times two makes four. So this first term becomes seven
times four. And with the second term, we’ve got
minus four times negative three all cubed. So negative three all cubed means
negative three times negative three times negative three. And negative three times negative
three is positive nine. And positive nine times negative
three is negative 27. So that is negative 27.

Now, we’ve got two lots of
multiplication and one lot of subtraction to do. And according to our order of
operations, we should do the multiplications first. So seven times four is 28. And we’re taking away four times
negative 27. Well, four times negative 27 is
negative 108. So if we take away a negative
number, then we’re adding that number. So 28 take away negative 108 is the
same as 28 plus 108, which is 136. So having these negative numbers to
plug into our equation. And having the squared and the
cubed terms in there. And then taking the negative of a
negative number here make that a much more tricky question.

Now, we’ve got to evaluate five 𝑚
divided by three 𝑛 minus two, given that 𝑚 is equal to 10 and 𝑛 is equal to
four.

Now, remember, the five up against
the 𝑚 means five times 𝑚, and the three up against the 𝑛 means three times
𝑛. So now, we’ve got the
expression. We can just substitute 10 in for 𝑚
and four in for 𝑛. Now, because we’ve got parentheses
or brackets here, we have to evaluate that first. Now, within those parentheses,
we’ve got three times four minus two. So we have to do the multiplication
and then we do the subtraction. Now, three times four is 12 and so
12 minus two is 10. Now, with this expression, we’ve
got multiplication and division. They’ve got the same level of
precedence, so we have to work from left to right. So first, we’ll do five times 10,
and we’ll take the result of that and then divide that by 10. So that’s 50 divided by 10, which
is just five.

Now, the next question is, evaluate
100 divided by two 𝑥 when 𝑥 is equal to 10.

Now, this question’s a little bit
tricky. The two 𝑥 together here implies
parentheses around them. That means they group together. So this is 100 divided by two 𝑥,
not just 100 divided by two times 𝑥. We’ll look at the difference
shortly. But when you’ve got the algebraic
version two 𝑥, it does imply parentheses. So you have to do that before you
do the division. So this means 100 divided by two
times 10. Well, two times 10 is 20. So this becomes 100 divided by 20,
which is equal to five. So the wrong version of that
would’ve been 100 divided by two times 𝑥.

Now, here we’ve got a division and
we’ve got a multiplication with equal precedence. So you’d think we would lo- wo-
work from left to right. So replacing 𝑥 with 10 gives us
100 divided by two times 10. Then 100 divided by two is 50. So that’s 50 times 10. And 50 times 10 is 500. So this is a bit of a trick
question. And my advice to you is if you’re
ever writing an expression that looks a bit like this. Rather than writing it that way,
write in one of these two ways to make it absolutely clear what you mean. Although there is this rule that
says algebraically that implies parentheses around the two. They have to happen first. That’s not so widely known. And you can understand why people
would make this sort of a mistake when doing that sort of question.

Lastly then, let’s evaluate five 𝑐
plus eight 𝑑 divided by four 𝑐𝑑 when 𝑐 is equal to five and 𝑑 is equal to
nine.

So again, we’ve got these algebraic
things. Five 𝑐 implies parentheses, eight
𝑑 implies parentheses, and four 𝑐𝑑 implies parentheses. It’s not just a simple case of
replacing 𝑐 with five and 𝑑 with nine and sticking multiplication signs between
those things. So that means five 𝑐 is five times
five, eight 𝑑 is eight times nine, and four 𝑐𝑑 is four times five times nine. So that’s the expression we end up
with. Let’s evaluate the inside of each
of those parentheses. Well, five times five is 25. Eight times nine are 72. And four times five times nine is
180.

Now, we’ve got an addition and a
division to do. And in our order of operations,
we’re gonna do the division first. So that means 25 plus 72 over
180. So that means 25 plus 72 over
180. Well, 72 over 180, 72 and 180 are
both divisible by nine. Nine is going to 72 eight times,
and nine is going to 180 20 times. So that becomes eight over 20. Now, fours go into eight twice and
fours go into 20 five times. So we’ve then got two-fifths. So our answer is 25 and
two-fifths.

Remember, it would’ve been very
tempting just to write the multiplication signs in there and substitute the numbers
in. And if we’d have followed that
process through, we’d have got a quite different answer of 835. So have a little think about how
you would write that expression, five 𝑐 plus eight 𝑑 divided by four 𝑐𝑑, to make
it absolutely clear what the correct intention was. Well, you could’ve put parentheses
around the four 𝑐𝑑 to make it absolutely clear that they should all stay
together. Or, you could’ve written five 𝑐
plus eight 𝑑 over four 𝑐𝑑 as an alternative way of making that absolutely
clear.