Lesson Video: Evaluating Algebraic Expressions | Nagwa Lesson Video: Evaluating Algebraic Expressions | Nagwa

Lesson Video: Evaluating Algebraic Expressions Mathematics

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).

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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.

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