Introduction to Algebra Word Expressions
We are mere operatives, empirics, and egotists until we learn to think in letters instead of figures.
—Oliver Wendell Holmes 1841–1935) American Physician and Writer
In this lesson, you'll learn how to translate word expressions into numerical and algebraic expressions.
You might not believe it, but real-life situations can be turned into algebra. In fact, many real-life situations that involve numbers are really algebra problems. Let's say it costs $5 to enter a carnival, and raffle tickets cost $2 each. The number of tickets bought by each person varies, so we can use a variable, x, to represent that number. The total spent by a person is equal to 2x + 5. If we know how many raffle tickets a person bought, we can evaluate the expression to find how much money that person spent in total.
How do we turn words into algebraic expressions? First, let's look at numerical expressions. Think about how you would describe "2 + 3" in words. You might say, Two plus three, two added to three, two increased by three, the sum of two and three, or the total of two and three. You might even think of other ways to describe 2 + 3. Each of these phrases contains a keyword that signals addition. In the first phrase, the word plus tells us that the numbers should be added. In the last phrase, the word total tells us that we will be adding.
For each of the four basic operations, there are keywords that can signal which operation will be used. The following chart lists some of those keywords and phrases.
Some words and phrases can signal more than one operation. For example, the word each might mean multiplication, as it did in the raffle ticket example at the beginning of the lesson. However, if we were told that a class reads 250 books and we are looking for how many books each student read, each would signal division
Let's start with some basic phrases. The total of three and four would be an addition expression: 3 + 4. The keyword total tells us to add three and four. The phrase the difference between ten and six is a subtraction phrase, because of the keyword difference. The difference between ten and six is 10 – 6.
Sometimes, the numbers given in a phrase appear in the opposite order that we will use them when we form our number sentence. This can happen with subtraction and division phrases, where the order of the numbers is important. The order of the addends in an addition sentence, or the order of the factors in a multiplication sentence, does not matter. The phrase one fewer than five is a subtraction phrase, but be careful. One fewer than five is 5 – 1, not 1 – 5.
Some phrases combine more tan one operation: ten more than five minus three can be written as 10 + 5 – 3 (or 5 – 3 + 10). As we learned in Lesson x 4, the order of operations is important, and it is just as important when forming number sentences from phrases. Seven less than eight times negative two is (8)(–2) – 7. We must show that 8 and –2 are multiplied, and that 7 is subtracted from that product.
Although multiplication comes before subtraction in the order of operations, a phrase might be written in such a way that subtraction must be performed first. The phrase five times the difference between eleven and four means that 4 must be subtracted from 11 before multiplication occurs. We must use parentheses to show that subtraction should be performed before multiplication: 5(11 – 4).
Tip:If you are writing a phrase as a number sentence, and you know that one operation must be performed before another, place that operation in parentheses. Even if the parentheses are unnecessary, it is never wrong to place parentheses around the operation that is to be performed first. |
Writing Phrases as Algebraic Expressions
Writing algebraic phrases as algebraic expressions is very similar to writing numerical phrases as numerical expressions. What is the difference between numerical phrases and algebraic phrases? Numerical phrases, like the ones we have seen so far in this lesson, contain only numbers whose values are given (such as five or ten). Algebraic phrases contain at least one unknown quantity. That unknown is usually referred to as a number, as in "five times a number." A second unknown is usually referred to as another number or a second number.
The variable x is usually used to represent "a number" in these expressions, although we could use any letter. Just as the phrase ten more than five is written as 10 + 5, the phrase ten more than a number is written as 10 + x.
When a phrase refers to two unknown values, we usually use x to represent the first number and y to represent the second number. Twice a number plus four times another number would be written as 2x + 4y.
Practice problems and solutions for these concepts at Algebra Word Expressions Practice Questions.
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