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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.

Did you ever encounter something like a + b = c and wonder how all these letters snuck into a math problem? In this lesson, we'll learn about variables, as well as other parts of equations, constants and coefficients.

Have you ever played poker? Sometimes when I play, I may look at my cards and be one card short of a good hand. Maybe I have three jacks. That's pretty good, but if I just had one more jack, I'd have four of a kind, which is great.

Unfortunately, a deck of cards only has four of each card, so the odds of getting all four jacks in your hand are small. But what if we were playing with wild cards? A wild card is a card that can be whatever you want it to be. It could be a jack for me or an ace for someone else. It's a card that has an undetermined value. This is just like a variable in algebra.

A **variable** is a symbol that represents an unknown number. It will be equal to a number, but you don't know what that number is yet. Until you figure it out, you use the symbol.

The variable you most often see is *x*. But a variable can really be any letter or symbol that doesn't otherwise have a number associated with it.

Think again about wild cards. A wild card has something printed on it, but its real meaning is determined by the other cards in your hand. In an algebraic equation, a variable's real meaning is determined by the things around it.

Before we go any further, let's learn about those other things around our variables. Here's an equation: 2*x* + 1 = 7.

What are 1 and 7? Numbers! Yes, they're numbers. But we also call them constants. A **constant** is a fixed quantity that cannot change. It's constant. Constants are our non-wild cards. Just like a 3 of hearts is always a 3 of hearts (supposing that card is never wild), and a 7 is always a 7. By the way, we can't use a number as a variable - since they're constants, that would be too confusing.

Back to our equation, there's also that 2. It has a special name. It's called a coefficient. A **coefficient** is a number that multiplies a variable. If you have 5*y*, your coefficient is 5. If you found out that *y* = 6, then 5*y* is 5*6, or 30.

You'll often be asked to solve equations with variables, like *x* - 2 = 5. But you should know that variables don't just exist for algebra quizzes. They have very practical purposes.

Let's say you're at your favorite local burger place. They sell double bacon cheeseburgers for $4 each. You want to get as many as you can, but you only have 13 dollars. Plus, if you're going to buy a bunch of double bacon cheeseburgers, you're also going to need a drink, which costs a dollar. How can you figure out how many burgers you can get? The number of burgers is your variable. Maybe it's 1, maybe it's 7. You don't know. So let's call it *x*.

Each burger costs $4, so 4*x* is the cost of a burger times the number of burgers. 4*x* + 1 is the cost of the burgers and the soda. That has to equal 13, so our equation is 4*x* + 1 = 13.

You just used a variable to help you solve a crucial problem and help you maximize your double bacon cheeseburger intake.

But how do you solve this equation? When you're trying to solve an equation that has a variable, you're trying to find out the value of that variable.

Let's first try a simpler example: *x* - 5 = 10. Our variable is *x*, so that's what we want to figure out. 5 and 10 are our constants. How do we figure out *x*? Well, our equation has two parts: *x* - 5 and 10. They're separated by an equals sign. In order to learn what *x* is we need to get it alone on one side of the equation. Here, we want to add 5 to both sides. This will get us *x* = 15. So what is *x*? It's 15!

We can check ourselves by putting 15 where *x* was in the original equation. So instead of *x* - 5 = 10, we have 15 - 5 = 10. Is 15 - 5 10? Yes!

Let's try one with a coefficient: 3*y* = 27. This time, our variable is *y*. Does it matter what letter it is? No! It could be *w* or *j* or that thing Prince was when he wasn't Prince for a while. It's just a symbol for an unknown number. What's a constant in this equation? 27. And our coefficient? 3.

To solve this one, we still want to get our variable by itself. That means we need to divide both sides by 3. That will get us *y* = 9. So our variable was standing in for 9. Let's check that. Does 3*9 = 27? Yes!

Okay, we built up our appetite. Let's figure out that burger equation. It was 4*x* + 1 = 13. Let's start by subtracting 1 from both sides to get 4*x* = 12. Then we divide both sides by 4. That gives us *x* = 3. Do you remember what *x* was symbolizing? Double bacon cheeseburgers, of course. So you can afford 3 double bacon cheeseburgers and a soda with your $13. Oh man, thank you algebra!

In summary, a variable is a symbol that represents an unknown number. Though it can be anything, you often see letters like *x* or *y* as variables in algebraic equations. The other parts of equations include constants, which are fixed quantities that cannot change, and coefficients, which are numbers that multiply a variable.

Once you have finished this lesson, you should be more confident in understanding what a variable is in an algebra problem and how to solve for it.

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ELM: CSU Math Study Guide16 chapters | 140 lessons

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- Evaluating Simple Algebraic Expressions 7:27
- The Commutative and Associative Properties and Algebraic Expressions 6:06
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