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Welcome to Algebra I: Solving Literal Equations

16 minutes

Hi, guys. Welcome to Algebra 1. Today's lesson's going to focus on solving literal equations. If you think back to pre-algebra, you had practice solving equations. You'll use that knowledge to get you through this one. You ready? Let's go.

(Describer) She uses a stylus on a screen.

Okay, before we get into literal equations, what they are, think back to linear equations. In pre-algebra, you probably solved something like this. You learned steps to get through this. You'd see, "solve for x." You've got 5x equals 20, so you think, when solving equations, the goal is to isolate the variable. You want to get the variable all alone. You have 5 times your variable, 5 times x. You have to do the opposite or the inverse operation-- in this case, of multiplying. The opposite of multiplying by 5 would be to divide by 5, so you'd divide both sides of your equation by 5.

(Describer) She writes.

On the left, the 5s would cancel out, so you'd be left with x equals-- and then 20 divided by 5 is 4. In pre-algebra you learn to solve one-step equations. We can solve them in one step. Okay? All right. Continuing back on memory lane, let's look at this.

(female describer) Title: Solve for x. 2/3x equals 4.

(Describer) Title: Solve for x. Two-thirds x equals four.

(teacher) You could solve this in one step or two depending on how you thought. I'll show you how to solve it in two. It probably alarmed you that it had a fraction, but we're not afraid. I'll show you how to get rid of it. Here I have 2/3x equals 4. I want to get rid of that fraction. That would probably make us happy. Let's move this a little bit. Look at the fraction separately. Consider its numerator; consider its denominator. We want to get rid of the denominator first. Two-thirds--if you thought about that like a division problem, it's 2 divided by 3, and we're multiplying that times x. I'm dividing by 3; thinking about this as 2 divided by 3. What's opposite of dividing by 3? Multiplying by 3, right? That's how you'll start to get rid of this fraction. I'll multiply the left side of my equation by 3. Put my multiplication symbol and a 3. Whatever you do to one side you do to the other. I multiply the right side by 3. On the left side, if you remember, your 3s cancel out. All you're left with is 2x, so 2x equals-- On my right side, I have 4 times 3. I know 4 times 3 is 12--there. We're at this point where we have 2x equals 12. We feel comfortable with that; it's similar to the last one. You look here-- the x2 is multiplying x. What's the opposite of multiplying by 2? Divide by 2, right? You would divide both sides by 2, your 2s would cancel out, so on the left you're left with x equals, and then 12 divided by 2 is 6. You'd see x equals 6. I know in pre-algebra you handled linear equations. Now look at this. Here our equation is A equals b times h, and solve for h. When we have a situation like this, we're solving a literal equation. What's the difference between a literal equation and a linear equation like the ones we were handling? For literal equations, your answer is not a value. It's not going to be x equals 4 or x equals 2. Your answer is some other algebraic expression, something else basically with variables still. Let me work you through this one. Here I'm solving for h, and I have a equals b times h because when I see terms together in algebra, multiplication is holding them together. I need to isolate h here; h is what I'm solving for. Pretend they're just numbers; you use the same process. I have b times h; what's the opposite of multiplying by b? Divide by b. We don't know what b is, but treat it the same way we treat a number like the other equations. We divide the right side by b. We're going to divide the left side by b. On the right side, "b"s cancel out and you're left with h. You have h, we have equals, and on the left we have a divided by b. You're done, for the most part; you've solved this. It makes us feel more comfortable if we see the variable on the left and the answer on the right. You might reverse this. I'll say our answer is a divided by b equals h, or h equals a divided by b. Either one is acceptable to write. You might feel comfortable seeing the variable on the left and the expression on the right, but they're both fine. You solved your first literal equation. You see what I mean-- this a divided by b. It's not a numerical value. We don't know values of a and b; they're variables. That makes literal equations different from linear equations. You may feel like you're not done, but as long as you solve for the variable, you're done. Let's look at example two. Here we're solving for r and we have v equals pi r squared h. A couple of things about this problem might have alarmed you. First of all, you probably noticed the pi. I'm sure you've learned that you can approximate pi at 3.14. It's a very long number that never ends and never repeats and goes on and on for infinity. Your calculator will usually take it 10 or 11 decimal places, but generally we round it at 3.14. The further you get in your math classes-- Algebra 1 and Geometry-- the further you go, you'll see "pi" more often than you'll see 3.14. When you see it, use it; don't feel you have to use 3.14. Go ahead and keep going with "pi." Another thing, you probably noticed we have an exponent. That adds to our steps when we're solving this problem, generally the same process we're doing every time. Let's jump in. Let me move this over so I have a little more space to work. I have v equals pi times r squared times h. I have the product of three terms on my right. I know I'm solving for r, so my goal is to isolate r. Right now it's r squared, so I'll work on getting that alone. If these are being multiplied together, I ask myself: What's the opposite of multiplying? Dividing. I want r squared to stay, but I want everything else to leave. I divide by what I want to get rid of. I divide by pi h-- by that product. Whatever I do to one side I do to the other, so I divide the left side by pi h. I've got that taken care of. On the right side your pis cancel, your "h"s cancel, so you're left with r squared. And on the left side, you're left with v divided by pi h. What is v? We have no idea. What is h? We don't know. That makes it literal, because you're working with a lot of variables in these kinds of equations. I have v divided by pi h equals r squared. I'm trying to solve for r, not r squared. I ask myself: What's the opposite of squaring a number? I need to undo the squaring. And it's square root. We take the square root of r squared and the square root of v divided by pi h. That allows us get at r, what we're trying to solve for. We'll take the square root of r squared and the square root of that quotient over there.

(Describer) She writes radicals.

The square root of r squared-- I know that's r. V divided by pi h-- that square root, I write it just like that because I don't have numerical values for v and h. The square root of v divided by pi h-- People are often more comfortable seeing the variable they were solving for on the left and then the answer on the right. So you could rearrange this. You could write this as r equals square root-- big square root-- v divided by pi h. Either are acceptable answers, just whichever one you feel more comfortable writing. Let's try another. See if you're feeling more comfortable with these. We have v equals 1/3bh. They're asking us to solve for b. We've got a fraction, but we practiced with that, so we're not alarmed, right? We're trying to get b, to solve for b. We'll feel better if we can get rid of the fraction. Consider the denominator first. My denominator is 3. If you think about that fraction, like 1 divided by 3, that will give you a hint of your inverse operation. If you're dividing by 3-- What's the opposite of dividing by 3? Multiplying by 3, right? I multiply the right side by 3 and the left side by 3. On the right side, that allows my 3s to cancel and I am left with 1bh. Because 1 is not going to change anything as a coefficient, I am going to write bh. Then I have 3 times v, so I have 3v. I'm not done because I'm trying to get b. I have 3v equals b times h, so h is multiplying times the b. So what's the opposite of multiplying by h? Dividing by h, right? Divide the right side by h; divide the left side by h. And what happens on the right is your "h"s cancel out. You're left with B. On the left, you have 3v divided by h. And you are done-- got through that. It took us about two steps. Okay? All right. It's your turn, guys. Press pause and take a few minutes and work through these problems. Notice the variable that I'm asking you to solve for. When you're ready to check your answers, press play.

(Describer) Number One: Solve for b. A equals one-half bh. Number Two: Solve for w. V equals Lwh.

(female describer) 1. Solve for b: A equals 1/2 bh 2. Solve for w: V equals lwh. Okay, let's see how you did. Get the pointer tool back. On the first one, you're solving for b. I got b equals 2a divided by h. On the second one, we were solving for w. I got w equals v divided by lh, okay? I'll show you how I got those; I'll show my work. Let's get the pen; okay, here we go.

(Describer) Number One:

I was solving for b. Let's move things out of the way and get room to work. Let's get that out of the way, and let's move this up. There we go. I want to get b by itself, so I look at my problem and see what's going on with b. I think I'll scoot that over more. First, I spot that fraction. Get that out of the way. Think about that fraction as 1 divided by 2. That hints that, to get rid of it-- or you'll hear people say "clear" their fraction-- I need to multiply both sides by 2. On the right, my 2s cancel out, and I'm going to write bh, because that 1 as a coefficient isn't changing anything. And then on the left I have 2 times a. That's 2a. I'm trying to solve for b. I have 2a equals b times h. The opposite of multiplying by h--divide by h. Divide the right side and the left side by h. On the right, your "h"s cancel out. So you're left with b. And on the right, 2a divided by h. You noticed, I reversed it when I wrote the answer. This one is acceptable also, okay? Let me show you how I got that next one. Let's move some things out of our way. Move that. This time they're connected, okay. I'll rewrite that so I won't get that too crowded. V equals l times w times h. I'm solving for w. What's going on here? I have v equals l times w times h. I want to get w by itself. It's being multiplied by l and h. What's the opposite of multiplying by l and h? Divide. I'll divide the right side by lh and then the left side. What I do to one side, I do to the other. On the right, my "l"s cancel, my "h"s cancel, and I just have w. On the left, v divided by lh. On the answers, I reversed these, but this is fine too. Whichever way you want to write it, okay?

(Describer) V divided by Lh equals w.

All right, guys. I hope you feel confident about solving literal equations. I hope to see soon for more Algebra 1. Bye.

(Describer) Accessibility provided by the US Department of Education.

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In this program, students learn that equations with several variables are called literal equations. These equations use variables to represent known values like distance, time, interest, and slope. Part of the "Welcome to Algebra I" series.

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Runtime: 16 minutes

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