An algebraic expression can be a single term that contains a variable, such as 3x, or many terms, such as 2x3+ 5x2– 7.
A term is a variable, a number, or a variable and a number that is multiplied, divided, or raised to an exponent. Terms can be added or subtracted.
A variable is a letter or symbol in an equation or an expression that holds the place of a number.
An algebraic expression is a single term or multiple terms on which one or more operations are performed and in which at least one vari- able is present.
A word problem can describe an algebraic expression in words and ask you to write that expression in numbers or variables, the way you normally see
it. First, let’s practice going in the opposite direction: writing algebraic expressions in words.
Example
What is 3x in words?
When a number, or constant, appears right next to a variable, that con- stant is multiplied by the variable. How can we describe the variable x? It is an unknown number, or simply “a number.” 3x is “three times a number.”
A constant is a real number, such as 8 or –1, and not a variable. Example
Describe x + 7 in words.
We still call x “a number,” so x + 7 is “seven more than a number” or “a number plus seven.”
Describing a single operation may not be hard, but when an expression contains multiple operations, we must be careful how we write the expres- sion in words, in order to preserve the correct order of operations and the correct interpretation.
Example
What is –7a + 5 in words?
This expression contains two terms, –7a and 5. However, –7 is multiplied by a before 5 is added, so we must be sure to keep the order of operations clear when we describe this expression. –7a is “the product of negative seven and a number,” so –7a + 5 is “the product of negative seven and a num- ber, added to 5.” If we had written “the product of negative seven and the sum of a number and 5,” we would have described the expression –7(a + 5), which is equal to –7a – 35, not –7a + 5, so the correct interpretation is just as impor- tant as the order of operations.
Now that we are comfortable turning algebraic expressions into words, we’re ready to handle algebraic word problems. Given an algebraic expres- sion in words, write the algebraic expression.
INSIDE TRACK WHEN CONVERTING Aphrase or a sentence into an algebraic expression, remember to check for keywords and backward phrases, just as you would when working with a word problem that doesn’t involve algebra.
Example
What is three less than a number?
This question is just like the questions we saw at the end of Chapter 5, only instead of writing a word problem solving an expression made up of only con- stants, we will be writing an expression that is a mix of constants and vari- ables. Just as with those problems, once we’ve converted the word problem into an algebra problem, we no longer need the eight-step process, because we no longer have a word problem.
Whenever you see a quantity in a word problem described as “a number” or “an unknown,” you know it’s an algebra problem, and you will need to use a variable to represent that number. Most people like to use “x” to represent unknown numbers, so we will do the same, but you could use any letter. If we use x to represent “a number” in this word problem, then “three less than a number” becomes “three less than x.” Rewrite any number words as num- bers. The expression is now “3 less than x.” As we learned in Chapter 1, the words less than signal subtraction, and as we learned in Chapter 4, these words can also be a backward phrase: 3 less than x is “x – 3.”
Example
Find seven divided by the sum of nine and a number.
The keywords divided and sum tell us that we will need to use division and addition. Convert the words that are numbers to real numbers, and rewrite
a number as “x”: Find 7 divided by the sum of 9 and x. Since 7 is divided by
a sum, we must write the sum before we can divide. The sum of 9 and x is 9 + x. The number 7 divided by that sum is (9 +7x).
CAUTION
WE MAY NEEDto add parentheses to an algebraic expression to preserve the correct order of operations. In the last example, if we had written 97+ x, our answer would have been incorrect. Division is per- formed before addition in the order of operations, so 97+ x is equal to “the sum of seven divided by nine and a number” and not “seven divided by the sum of nine and a number.” Always place parentheses around the operation you want to be performed first. Even if that oper- ation would have been performed first anyway, it would not be incor- rect to place the parentheses. If we wanted to write the expression “the sum of seven divided by nine and a number,” we could write either 97+ x or (79) + x, and both expressions would be correct.
PRACTICE L AP
DIRECTIONS: Write each of the following phrases as an algebraic expression.
1.eight more than a number
2.a number divided by six
3.eleven times the sum of a number and five
4.the quotient of a number and the difference between nineteen and two
5.twice the product of two different numbers
6.one more than the difference between six and a number
7. the square of a number, minus five times that same number
PACE YOURSELF WRITE AN ALGEBRAICexpression to represent what your height will be in five years. What numbers or variables are in your expres- sion? How could you evaluate your expression five years from now?