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ELM: CSU Math Study Guide16 chapters | 140 lessons

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

In this lesson, we'll learn how to evaluate algebraic expressions, which involves substituting numbers for variables and following the order of operations. By the end of the lesson, you'll be an algebraic expression expert.

We should always remember that algebra has a basis in real life. When we have all these variables and exponents and things, that's easy to forget. But the purpose of algebra is to help us find answers to real situations.

For example, let's say that you have a part-time job as a dog walker. You earn $15 each time you walk a dog. If *d* is the number of dogs you walk, your total income is $15*d*, or $15 times the number of dogs. That's an algebraic expression. And what if you had 6 walks scheduled this week? How much will you earn? That's where we get to evaluating algebraic expressions, our topic for this lesson.

**Evaluating algebraic expressions** is when you substitute a number for each variable and then solve the expression. These types of problems typically look like this: Evaluate 15*d* when *d* = 6.

So this is our dog walking scenario. And you can see there are two parts to the question. First, there's an algebraic expression. This is a mathematical sentence of sorts that contains one or more variables. Here, it's 15*d*. Second, there's a sample value for the variable. In this case, we have *d* = 6.

To solve this problem, we take the value and plug it into our expression. So we'd replace the *d* in 15*d* with a 6. We complete the problem by multiplying 15 times 6, which gets us 90. That's our answer! You'll earn $90 this week - plus you'll get lots of exercise, which is kind of its own payment, right? Well, okay, the money's good, too.

Evaluating algebraic expressions can be pretty straightforward. Whether you have one variable or ten, it's really just plugging them in and solving. Usually, the trickiest part is remembering the order of operations. The **order of operations** is a system that determines which procedures occur first in an expression.

As you'll recall, the mnemonic **PEMDAS** gives us our order. PEMDAS stands for *parentheses, exponents, multiplication, division, addition and subtraction*. You might remember PEMDAS with the phrase *Please Excuse My Dear Aunt Sally*. This mnemonic makes inoffensive Aunt Sallys everywhere cringe. Offensive Aunt Sallys don't care what you say about them.

But why do we care about Aunt Sally? Well, if we had 4 - 2 * 3, would we do 4 - 2, which gets us 2, then multiply that by 3, which gets us 6? Or would we multiply 2 * 3, which gets us 6, then subtract that from 4, which gets us -2? Those are two totally different answers. PEMDAS tells us multiplication precedes subtraction, so -2 is the correct answer.

Also, remember that multiplication and division are linked, as are addition and subtraction. Or, 'my dear' is a package deal, as is 'Aunt Sally.'

Ready to try some practice problems? Almost. First, a warning. It's critical that you always use parentheses. Did you ever go on a carnival ride where they tell you to keep your arms inside the ride? Parentheses are the ride, and minus signs are the numbers' arms. They even kind of look like arms, which is a bonus.

Here's why this matters. Let's say you have *x*2 - 3 and you want to evaluate it when *x* = 2. Okay, no problem. That's 22 - 3. Exponents are first, so I square the 2 and now I have 4 - 3. That's easy: 1. You might be thinking, 'I don't need parentheses. I solved that problem without them just fine.'

Okay, what if *x* = -2? So we have -22 - 3. Again, I square the 2 and I get -4 - 3. That will be -7. And that's wrong. You should have squared not just 2 but -2. Then you'd get 4 - 3 and again get 1.

That would've been easy to remember if you'd written the expression as (-2)2 - 3. So consider this a safety reminder: Always use parentheses. Don't let your numbers lose their arms.

Okay, let's practice evaluating some algebraic expressions. Let's start simple: Evaluate *y* - 2 when *y* = 10. Just plug 10 in for *y*: 10 - 2. Then solve. Our answer is 8. That means that when *y* = 10, *y* - 2 equals 8.

Here's another: Evaluate *a*2 + 5 when *a* = 3. If we plug 3 in, we get 32 + 5. Now, remember PEMDAS. Exponents come before addition. So we first square the 3 to get 9 + 5. And 9 + 5 is 14. So when *a* = 3, *a*2 + 5 = 14.

Let's get more complicated: Evaluate 4*xy*3 - 12 when *x* = -2 and *y* = 3. Okay, don't worry. We can handle this. The first step is to plug in our *x* and *y* values. We get 4(-2)(3)3 - 12. Never forget those parentheses. Keep your arms and minus signs inside the ride.

Now, it's 'please excuse my dear Aunt Sally' time. We'll start with the exponent. 33 is what? 3 * 3 = 9 and 9 * 3 = 27. So now we have 4(-2)(27) - 12. The next step is multiplication. 4 * (-2) = -8. And -8 * 27 = -216. That gives us -216 - 12. That's -228. That's our answer!

I think you can handle one that's even harder: Evaluate:

when *a* = 4 and *b* = -2. Okay, that's a big messy one. But just follow these steps. First plug in our *a* and *b* values. We get:

And what about Aunt Sally? Okay, parentheses are first, so let's tackle that stuff inside the parentheses. 42 is 16 and -23 is... 8 or -8? It's -8. So we have 16 + (-8), which is 16 - 8, or 8.

Let's look at where we're at.

Let's simplify that to 16/(-8). And that'll just be -2. That big messy problem? It's just -2. Not bad! Always remember, if you can please excuse my dear Aunt Sally, you can evaluate algebraic expressions.

In summary, evaluating algebraic expressions is when you substitute a number for each variable and then solve the expression. Usually, the trickiest part is remembering the order of operations. For this, we use PEMDAS. This stands for *parentheses, exponents, multiplication, division, addition and subtraction*. You can remember the acronym with the phrase *Please Excuse My Dear Aunt Sally*.

When this lesson is done, you should be able to confidently evaluate simple algebraic expressions using the order of operations.

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ELM: CSU Math Study Guide16 chapters | 140 lessons

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