In this video, we’re gonna look at how to evaluate algebraic expressions,
given specific values of the 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 ten.
So we’re told that 𝑥 is equal to ten and wherever we see
𝑥 in our expression, we have to replace that with ten. But we have
to think quite carefully. We don’t just literally write ten because that would
then be seven hundred and ten 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 ten, that means seven times ten 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 ten is seventy. So that’s seventy plus
five, which gives us an answer of seventy-five.
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 thirty-two. And a negative times a negative makes a positive, so our answer is thirty-two.
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 operations, 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 forty-five. 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 forty-five add six is fifty-one.
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
twenty-seven. So that is negative twenty-seven. 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 twenty-eight. And we’re taking away four times negative twenty-seven. Well
four times negative twenty-seven is negative a hundred and eight. So if we take away a negative number, then we’re adding that number. So
twenty-eight take away negative a hundred and eight is the same as
twenty-eight plus a hundred and eight, which is a hundred and thirty-six. 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 ten 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
ten 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 twelve and so twelve minus two is
ten. 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 ten, and we’ll take the result of that and then divide that by
ten. So that’s fifty divided by ten, which is just five.
Now the next question is: Evaluate a hundred divided by two 𝑥
when 𝑥 is equal to ten.
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 a hundred
divided by two 𝑥, not just a hundred 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 a hundred divided by two times ten. Well two
times ten is twenty. So this becomes a hundred divided by twenty, which is equal to five.
So the wrong version of that would’ve been a hundred 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
ten, gives us a hundred divided by two times ten. Then a
hundred divided by two is fifty. So that’s fifty times ten. And fifty times ten is five
hundred. 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 a 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 twenty-five. Eight times nine are
seventy-two and four times five times nine is one hundred and eighty. 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 twenty-five plus seventy-two over a hundred and
eighty. So that means twenty-five plus seventy-two over one hundred and
eighty. Well seventy-two over a hundred and eighty,
seventy-two and a hundred and eighty are both divisible by
nine. Nine is going to seventy-two
eight times, and nine is going to a hundred and eighty
twenty times. So that becomes eight over twenty. Now
four is going to eight twice and four is going to
five times. So we’ve then got two-fifths. So our answer is twenty-five 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 eight hundred and thirty-five. 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 always stay together. Or, you could’ve written five 𝑐 plus eight 𝑑 over four 𝑐𝑑 as an
alternative way of making that absolutely clear.