2.1 The Language of Algebra

Introduction

In English, it's not uncommon for a word to have more than one meaning. Sometimes a double meaning can make you smile, as with the classic joke "Why was the math book sad? Because it had too many problems." However, multiple meanings can also leave you confused, like when a friend asks if you saw a green tank on your way to school. Are they asking about a large container, or a military vehicle?

In mathematics, we try our best to use precise language with agreed-upon meanings for specific words and phrases to avoid confusion. Spending time early in the course to become familiar with this terminology will make the rest of the term go much more smoothly.

Constants & Variables

As you might be able to guess, a constant is something that does not change; while a variable is a quantity whose value will change.

For example, the distance from my home to work is 7 miles. That distance doesn't change. It is constant. The time it takes to travel from home to work is variable, however. It can change from one day to the next based on the speed of traffic.

In algebra letters can be used to represent constants and variables. Generally speaking, we usually use letters and the beginning of the alphabet \(a\), \(b\), \(c\) to represent constants and letters toward the end of the alphabet \(x\), \(y\), \(z\) to represent variables.

That rule is not set in stone. Sometimes it makes sense to use a letter that matches the context of a particular quantity. For example, we could use \(d\) to represent the constant distance from your home to work and \(t\) to represent the variable time it takes to travel.

Algebraic Expression & Equations

Expressions and equations are two important concepts in algebra that are related, but different, and easy to confuse.

When constants and variables are combined together using operations (like addition, multiplication, etc.) we have created an expression. An equation, on the other hand, is a statement about the equality of two expressions. An equation will always have an equal sign \(=\) in it. For example, \(13+5\) is an expression whereas \(13+5 = 18\) is an equation. Here are a few more examples to help you see the difference.

Note that expressions and equations can both involve a combination of letters and numbers, which is perhaps why they are easily mistaken for each other. Again, the key difference is that equations always have an equal sign but expressions never do. For example, \(2x+5\) and \(11x\) are both expressions while \(2x+5 = 11x\) is an equation.

Evaluating and Solving

It's important to recognize whether you are working with an expression or an equation because we do different things with them.

Because equations have an equal sign, we can solve for the value of a variable that makes an equation true. For instance, the equation \(x+1 = 4\) can be solved by realizing that if the variable \(x = 3\) then both sides of the equation would have the same value of \(4\). In this case, we would say that \(x=3\) is the solution to the equation. In Chapters 3 and 4 we will explore a number of ways to systematically solve equations.

Since expressions do not involve an equal sign, we cannot solve them. We can, however, evaluate expressions when we are given specific values. If an expression contains variables, then we will need to replace those letters with numbers before finding the value represented by the expression. For example, if tacos cost \(\$2\) and burritos are \(\$5\), then the expression \(2t + 5b\) could tell us how much was spent on an order of tacos and burritos, but only if we knew how many of each were purchased. If someone orders one taco and two burritos, then substituting \(t=1\) and \(b=2\) gives

\begin{align} 2t + 5b &= 2 \cdot 1 + 5 \cdot 2 \\ &= 2 + 10 \\ &= 12 \end{align}

and we now know our expression has a value of \(\$12\).

Equivalent Expressions

Equivalent expressions represent two different ways of calculating the same value. You are likely already familiar with some basic rules that produce equivalent expressions, even if you don't know their names, and we will discover others later in this chapter.

Suppose, for instance, that two employees are planning a company event. They have \(12\) tables and each table seats \(8\) people. One employee uses the expression \(12\cdot 8\) to calculate the number of chairs needed. The other calculates the total number of chairs with the expression \(8 \cdot 12.\) These both produce the same result of \(96\) so they are equivalent.

This is an example of the commutative property for multiplication, which says that \(a \cdot b\) and \(b \cdot a\) always produce the same result. And since \(a+b\) and \(b+a\) always produce the same result, there is also a commutative property for addition.

Notice that the commutative property only works with addition or multiplication, it does not apply to subtraction or division. You can switch the order of the numbers you are adding or multiplying and still get the same answer, but if you change the order of subtraction or division the final value will not be the same.

Combining Like Terms

Expressions can be simple or complicated, depending on the number of constants and variables and the number of operations needed to evaluate them. For example, \(3x\) is a fairly simple expression, while \(8x^2-11x+4x^2+5x\) is a bit more complicated.

When given a complicated expression we will often try to write it in a simpler format that is equivalent to the original expression and easier to work with. Throughout the rest of Chapter 2, we will learn a number of techniques for simplifying expressions. Here we will focus on simplifying by combining like terms.

Before we talk about combining like terms, let's define a few more important words. When we add or subtract quantities, each individual quantity is called a term. The expression \(2+7\) involves the two terms \(2\) and \(7\) since those are the parts being added together.

When we multiply quantities together, each individual quantity is called a factor. The expression \(3(-5)\) involves two factors, \(3\) and \(-5\), since those are the values being multiplied together.

When a variable is multiplied by a constant, we call that constant the coefficient of the variable. In the expression \(5x+3y\), \(5\) is the coefficient of \(x\) and \(3\) is the coefficient of \(y\).

Two terms are considered like terms if they have the exact same variables raised to the exact same powers.

• The expressions \(32x\) and \(3x\) are like terms because they both involve the variable \(x\).
• The expressions \(-5.6x^2y\) and \(31x^2y\) are like terms because they are both multiples of \(x^2y\).
• The expressions \(21x^2\) and \(7x\) are not like terms. While they involve the same variable, they do not have the same powers.

Using this process we are able to return to the earlier examples and combine the like terms.

When there are several like terms, the commutative property can be used to move like terms next to each other so that they are easier to combine. This is the case with the more complicated example we saw at the start of this subsection.

\begin{align} 8x^2-11x+4x^2+5x &= 8x^2+4x^2-11x+5x \\ &= (8+4)x^2 + (-11+5)x \\ &= 12x^2 - 6x \end{align}