Video: Evaluating Algebraic Expressions

Tim Burnham

Through a series of increasingly difficult examples, we show you how to evaluate algebraic expressions, given specific values of the variables they contain (e.g., evaluate 5𝑎 − 2𝑏 when 𝑎 = 9 and 𝑏 = −3 and evaluate (5𝑐 + 8𝑑)/4𝑐𝑑 when 𝑐 = 5 and 𝑑 = 9).


Video Transcript

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 twenty 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.

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