Lesson Video: The Language of Algebra | Nagwa Lesson Video: The Language of Algebra | Nagwa

# Lesson Video: The Language of Algebra Mathematics

In this video, we will learn how to identify variables and terms, and distinguish between expressions, equations, formulae, and identities.

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### Video Transcript

In this video, we’re going to look at the different words that we use in algebra. We’ll identify terms, variables, and constants and see how we can tell the difference between expressions, equations, formulas, and identities. Let’s begin by looking at some of the basic building blocks that we begin with in algebra. Let’s start with defining a constant. This is simply a number on its own. We call it a constant because it never changes. For example, if we had three, that would always be the value of three. Minus one-half and 100 are also two examples of a constant.

The word variable means that this is a symbol for a number that we don’t yet know. Examples of variables would be 𝑥, 𝑦, or 𝐴. You might wonder why we use a variable if a value doesn’t vary. For example, if we had 𝑥 plus three equals five, then we know that 𝑥 would be equal to two, which doesn’t vary. But in the case of something like 𝑦 equals two 𝑥 plus three, then both the 𝑥 and 𝑦 would vary. We can then start to put our constants and variables together to form terms. A term is a single constant or variable or constants and variables multiplied together. For example, seven would be a term. It’s a constant as well, but it would be a term. Three 𝑥 and 𝑥𝑦 squared would also be two more examples of terms.

We can then put terms together to form expressions. For example, five plus two 𝑥 would be an expression, so would three 𝑥 minus 𝑦 squared plus five. The very important thing to remember about expressions is that they do not contain an equals to symbol. We can move from expressions to equations. An equation shows that two expressions are equal and, therefore, this will contain an equals to symbol. An example of an equation is seven plus two equals nine. Two more would be three 𝑥 equals 30 and 𝑦 equals seven 𝑥 squared minus five. When we talk about solving an equation, that means we’re finding the value of the unknowns or the variables.

Next, we have formulas. These are equalities or equations that show how the value of a quantity depends on the value of other quantities. A formula will always be an equation. An example of a formula would be 𝐹 equals 𝑚𝑎, so is 𝑉 equals 𝑙 times 𝑤 times ℎ, which we might recognize as the formula for the volume of a rectangular prism. There’s just one other definition to look at, and that’s for the word identity. An identity is an equality that is always true; that is, it is true whatever values the variables take. Often, we’ll see an identity written with the triple bar symbol rather than the equal symbol. So what examples could we give of an equality that’s always true?

Well, we could have something like this: 𝑥 plus 𝑦 multiplied by 𝑥 plus 𝑦 is equivalent to 𝑥 squared plus two 𝑥𝑦 plus 𝑦 squared. We might recognize that the right-hand side is the expansion of the left-hand side. Another example of an identity is 𝑎 over two equals 0.5𝑎. Even though this one doesn’t have the identity symbol, we know that it would still be an identity because that would always be true regardless of the values of 𝑎. Sometimes it’s difficult to tell the difference between an equation and an identity. Let’s for a moment consider these two expressions. We have two times three 𝑥 plus four, and we have six 𝑥 plus eight.

So, are these two expressions equal? Do they have the same value? Well, yes they do because we know that the right-hand side is the expansion of the left-hand side. If we wanted to solve this by trial and improvement, then any value of 𝑥 would work. For example, if we choose the value of 𝑥 equals one, on the left-hand side, we’d have two times three times one plus four. And on the right-hand side, we’d have six times one plus eight. Simplifying the left-hand side and the right-hand side, we would say that these two values are equal. Even if we chose a value like 𝑥 equals 100, we’d still have the left-hand side equal to the right-hand side. So let’s consider the two expressions two times three 𝑥 plus four and five 𝑥 plus 10.

Are these two expressions equal? Well, we don’t actually know. However, if we were told that both of these were equal, we could then solve it to find the value of 𝑥. Expanding the parentheses and then rearranging, we would find that 𝑥 is equal to two. So, what’s the difference between these two statements? Well, in our first one, we have two times three 𝑥 plus four is equal to six 𝑥 plus eight. We find that that equation is always true. This means that this must be an identity. In our second one, we have two expressions that we’re told are equal, which means that this must be an equation. We can indicate an identity with three bars, but we don’t always need to. Now that we’ve covered some of the main words that we use in algebra, let’s have a look at a few questions.

Fill in the blank. What is an equation that is always true, no matter what values are chosen.

Let’s begin by recalling what we mean by an equation. The important thing about an equation is that it has the equals to symbol. It will have an expression on the left-hand side and one on the right-hand side. So we could, for example, have an equation like six 𝑥 plus one equals 13. So what do we mean by an equation that’s always true no matter what values are chosen? We could see in the equation that we’ve chosen that this wouldn’t always be true. We could solve this to see that the value of 𝑥 must be two, as six times two plus one would give us 13. So this would not be a good example of an equation that’s always true.

If we wrote an equation like 0.1𝑥 equals 𝑥 over 10, then this would actually always be true. We could choose any value of 𝑥, and the left-hand side would always be equal to the right-hand side. There is, in fact, a special word for an equation that’s always true no matter what values are chosen, and that’s identity. Therefore, our answer would be an identity.

Identify whether five 𝑎 minus three equals 22 is an expression, an equation, an identity, a formula, or an inequality.

In order to answer this question, we’ll need to recall what it means to have an expression, an equation, an identity, a formula, or an inequality. Let’s begin with thinking what it means to be an expression. An expression is a collection of terms. So in this question, five 𝑎 minus three would be an expression, and 22 would be an expression. As the whole statement, five 𝑎 minus three equals 22, is not an expression on its own, then we could say that this is not an expression. There is, in fact, a mathematical word for when we have two expressions linked by an equal sign, and that word is an equation.

At this point, it looks as though our statement is an equation. But let’s see if any of the other three words would apply. Let’s take the word inequality. An inequality would, in fact, have to have one of the inequality symbols, for example, greater than, greater than or equal to, less than, or less than or equal to. As the statement that we’re looking at has an equal sign, then this would not be an inequality. So what do we mean when we talk about an identity? We can recall that an identity is always true no matter what values the variables take. So, if we look at our statement, will this be true no matter what the value of 𝑎 is?

Let’s say we chose the value of 𝑎 to be 100. On the left-hand side, we’d have five times 100 minus three, which is 497, and that is not equal to 22. We, therefore, cannot say that the statement five 𝑎 minus three equals 22 is always true. We have to say that this is not an identity. So finally, let’s consider if this would be a formula. We can recall that a formula is an equality that shows how the value of a quantity depends on the value of other quantities. It can be difficult to tell the difference between formulas and equations because every formula will be an equation. We can think of it somewhat informally by thinking that a formula is like a useful equation.

For example, we might have area is equal to length times the width, or we might have 𝐹 equals 𝑚𝑎. Usually in formulas, we’ll have at least two variables. In the statement five 𝑎 minus three equals 22, we just have one variable. So we don’t really see how the value would change based on other variables. So this really just leaves us with one answer option. We would say that five 𝑎 minus three equals 22 is an equation. And that’s because it has that equals to sign linking two expressions.

Identify whether 𝑥 minus 𝑦 all squared equals 𝑥 squared minus two 𝑥𝑦 plus 𝑦 squared is an expression, an identity, a formula, or an inequality.

Let’s have a look at this statement 𝑥 minus 𝑦 all squared equals 𝑥 squared minus two 𝑥𝑦 plus 𝑦 squared. If we take a look at the left-hand side, this would be composed of the set of parentheses 𝑥 minus 𝑦 times 𝑥 minus 𝑦. We could expand these parentheses, for example, using the FOIL method. We would get 𝑥 squared minus two 𝑥𝑦 plus 𝑦 squared, which means that we have a left-hand side which is equal to the right-hand side. So this will be true for any values of 𝑥 or 𝑦. It doesn’t matter what values we choose; the left-hand side will always be equal to the right-hand side. The word we use for a statement like this would be an identity.

Sometimes when we have an identity, instead of having this equals to symbol, we could use the identity symbol. But we don’t always need to write it, so we can’t use it really as an indicator if something is an identity or not. If we look at the other options we’re given, we can say that our statement is not an expression, as the expression would be the collection of terms that we have either on the left-hand side or on the right-hand side. A formula is a type of equation that we might use to help us find the value of a variable, given how another variable has changed. An example of a formula might be, for example, the circumference of a circle is equal to 𝜋 times the diameter. This isn’t quite what we have in our statement.

Finally, if we had an inequality, we’d be looking for one of the symbols, for example, less than or greater than or equal to. And we haven’t got one of these in our statement. We can then give our answer that 𝑥 minus 𝑦 squared equals 𝑥 squared minus two 𝑥𝑦 plus 𝑦 squared is an identity.

Let’s look at one final question.

Identify whether four 𝑥 minus five 𝑦 is an expression, an equation, an identity, a formula, or an inequality.

Let’s take a look at what we have here: four 𝑥 minus five 𝑦. If we were answering an algebraic problem and we had this statement, four 𝑥 minus five 𝑦, we couldn’t do very much with this. We might be able to collect any like terms, but we couldn’t solve it. We can’t solve it because we’re missing something. We’re missing an equals sign. What we have here with four 𝑥 minus five 𝑦 is two separate terms linked by the subtraction. The word that we use for this is an expression.

If we look at an equation, we can remember that this needs an equals to symbol, and so do identities and formulas. So straightaway, we could say that four 𝑥 minus five 𝑦 is not an equation, an identity, or a formula. An inequality needs to have an inequality symbol, for example, less than or greater than or equal to. So four 𝑥 minus five 𝑦 is not an inequality. We can give our answer that four 𝑥 minus five 𝑦 is an expression.

We can now recap some of the key points of this video. Firstly, we saw two words, constants and variables, which collectively make up different terms, for example, three 𝑥 or 𝑦 squared. Expressions contain constants and/or variables but, importantly, have no equals to sign. Next, equations state that two expressions are equal. And note that these will always have the equals to sign. Formulas are equalities or equations that show how the value of a quantity depends on the values of other quantities.

And finally, identities are equalities that are always true whatever the values the variables take. We might often see identities written with the identity symbol, with three bars, instead of using the equals to sign. It’s important that we’re confident with the meaning of all of these different words as this will help us with our mathematics.

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