Writing Algebraic Expressions
Many verbal phrases occur repeatedly in applications. The following list of
some frequently occurring verbal phrases and their translations into algebraic
expressions will help you translate words into algebra.
Translating Words into Algebra

Verbal Phrase 
Algebraic Expression 
Addition: 
The sum of a number and 8 Five is added to
a number
Two more than a number
A number increased by 3 
x + 8 x + 5
x + 2
x + 3 
Subtraction: 
Four is subtracted from a number Three less
than a number
The difference between 7 and a number
Some number decreased by 2
A number less than 5 
x  4 x  3
7  x
x  2
x  5 
Multiplication: 
The product of 5 and a number Seven times a
number
Twice a number
Onehalf of a number 
5x 7x
2x

Division: 
The ratio of a number to 6 The quotient of
5 and a number
Three divided by some number 

More than one operation can be combined in a single
expression. For example, 7 less than twice a number is written as 2x  7.
Solving Problems
We will now see how algebraic expressions can be used to form
an equation. If the equation correctly models a problem, then we may be able to
solve the equation to get the solution to the problem. Some problems in this
section could be solved without using algebra. However, the purpose of this
section is to gain experience in setting up equations and using algebra to solve
problems. We will show a complete solution to each problem so that you can gain
the experience needed to solve more complex problems. We begin with a simple
number problem.
Example 1
A number problem
The sum of three consecutive integers is 228. Find the
integers.
Solution
We first represent the unknown quantities with variables. The
unknown quantities are the three consecutive integers. Let

x 
= the first integer, 

x + 1 
= the second integer, 
and 
x + 2 
= the third integer. 
Since the sum of these three expressions for the consecutive
integers is 228, we can write the following equation and solve it
x + (x + 1) + (x + 2) 
= 228 
The sum of the integers is 228. 
3x + 3 3x
x 
= 228 = 225
= 75 

x + 1 
= 76 
Indentify the other unknown quantities. 
x + 2 
= 77 

To verify that these values are the correct integers, we
compute
75 + 76 + 77 = 228.
The three consecutive integers that have a sum of 228 are 75,
76, and 77.
