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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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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.

When we need to simplify an expression, how do we know if we should distribute first or add first? In this lesson, we'll look at when this matters and how to make the best decision.

When you get ready to go out in the morning, do you put your socks on first or your shoes? Socks, right? What about your pants or your shirt? You probably have a habit of doing one or the other first, but it doesn't really matter, does it?

What about simplifying algebraic expressions? We know the **distributive property** tells us that *a*(*b* + *c*) = *ab* + *ac*. But what if we have 3(2 + 4)? Should we do 3 * 2, which is 6, then 3 * 4, which is 12, to get 6 + 12, which is 18? Or should we do 2 + 4, which is 6, then 3 * 6, which is, again, 18?

This is a shirt vs. pants kind of example. It really doesn't matter. We'll get the same answer, 18, just as we'll still end up with our shirt and pants on no matter what we do first.

How do we know if we should add first or distribute first? There are a few factors to consider.

Let's start with the most crucial question: can you add first? We're looking to see if we can combine like terms. If the answer is no, well, then it's time to distribute first.

If you have different variables, like 2(3*x* + 2*y*), you can't add first. You can't combine 3*x* and 2*y*. That's like trying to put your belt on before your pants. Likewise, if you have something like 5(1 + 2*x*), you have one term with a variable, 2*x*, and one term that's just a constant with no variable, 1. You can't combine those either. Sorry, belt, you have to wait.

Next question: is adding or distributing first easier? This is subjective. Is it easier to pour the milk after putting the cereal in the bowl? Yes. Then you know how much milk to add. Can you start with the milk? I guess so, though that would be kind of weird.

Here's an expression: 2*x*(1.7 + 3.3). If we distribute first, we do 2*x* * 1.7, which is 3.4*x*, and 2*x* * 3.3, which is 6.6*x*. 3.4*x* + 6.6*x* = 10*x*. That's not so bad. But we had to do two steps with decimals. If we added first, we'd do 1.7 + 3.3, which is 5. Then 2*x* * 5 is 10*x*. That's easier. Adding first is often easier with decimals.

What about fractions? Do we go milk or cereal first? Look at 3*y*(1/4 + 1/2). If we distribute first, it's 3*y* * 1/4, which is 3*y*/4, then 3*y* * 1/2, which is 3*y*/2. Now we have 3*y*/4 + 3*y*/2. We need to get common denominators, so we multiply the second fraction by 2/2 to get 6*y*/4. Now we can add them to get 9*y*/4. Whew.

What if we added first? 1/4 + 1/2. Ok, that's 1/4 + 2/4, which is 3/4. 3/4 * 3*y*? 9*y*/4. Adding first is often easier with fractions, too.

This isn't always the case, though. Look at this one: 9*x*(2/3 + 1/9). We can make 2/3 into 6/9, so 6/9 + 1/9 is 7/9. Then we can multiply that by 9*x* to get 63*x*/9. That simplifies to 7*x*. But what if we distribute first? We do 9*x* * 2/3, which is 18*x*/3, or 6*x*. Then 9*x* * 1/9, which is 9*x*/9, or just *x*. 6*x* + *x* is 7*x*. Distributing first really wasn't any harder, was it?

It's ultimately a judgment call. Know that you'll get the correct answer no matter what you do. Just take a look at the problem and decide if one method will give you numbers you'd prefer to work with. In that last example, maybe you'd rather not get that 63*x*/9, which is slightly more complicated than handling the smaller numbers you get with distributing first.

Imagine you need to put on a tie for work. If you've never done this before - maybe you've lived in clip-on territory for a while - then it can be tough to master without practice. When we simplify expressions using distribution, practice is key.

Let's practice. Here's one: 35*x*(7 + 9). Ok, what should we do here? Can we add first? Yes. Should we? Well, if we don't, we'll have to add two big numbers once we distribute that 35. I say we add first. 7 + 9 = 16. 35*x* * 16 = 560*x*. Adding first was the way to go.

How about this one? 12*x*(1/12 + 4/3) This one has fractions, so sometimes adding first is easier. But notice that it might be easier here to distribute first, since the 12 will eliminate the fractions. Let's try it. 12*x* * 1/12 = 12*x*/12, or *x*, and 12*x* * 4/3 = 48*x*/3, or 16*x*. *x* + 16*x* = 17*x*.

If this were tying ties, we'd almost be a master. No more clip-ons for us! Let's do one more: 5*x*(1.9 + 2.3 + 4.8). Now we have decimals. Remember, look to add decimals first. That's usually easier. Fortunately, adding first makes this one very neat. 1.9 + 2.3 + 4.8 = 9. We just got rid of our decimals, and 5*x* * 9 = 45*x*. That's our answer!

To summarize, when we simplify problems involving distribution, we may need to decide whether we add first or distribute first. The distributive property tells us that *a*(*b* + *c*) = *ab* + *ac*, but, when possible, we can add *b* and *c* first, then multiply that sum by *a*.

There are a few good questions to ask before you decide. First, can you add first? If we can't combine like terms, then it's a moot point.

If addition is possible, remember that decimals and fractions are often, though not always, easier to add first. Try to determine which method will give you simpler numbers to work with, then distribute like an expert!

After you've reviewed this video lesson, you should be able to:

- Recall the distributive property
- Explain when it is easier to add before distributing

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- Why Do We Distribute in Algebra? - Explanation & Examples 5:39
- Distributing First vs. Adding First: Differences & Examples 6:44
- Distributing Algebraic Expressions with Numbers and Variables 7:57
- Changing Negative Exponents to Fractions 6:24
- Working With Fractional Powers 6:38
- Distribution of More Than One Term in Algebra 6:12
- Go to High School Algebra: Algebraic Distribution

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