Introduction

A variable is any letter or symbol that can be replaced by a number. For example, the letters x, \quad y, \quad z, \quad a, \quad b, \quad c, \quad k, \quad m,\quad n,\quad p, \quad q,\quad r, \quad \ldots

are commonly used as variables. But in general, we can use any letter we like.

An algebraic expression is a combination of variables with mathematical operations, like addition (+) , subtraction (-) , multiplication (\times) , and division (\div). So x+2,\qquad 6-y,\qquad 2\times p + 1,\qquad q\div r, are all examples of algebraic expressions.

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Example: Identifying Variables

Which of the following is a variable?

  1. x
  2. x+1
  3. 1
  4. x+1=2
EXPLANATION

Let's look at each one in turn.

  1. x is a variable because it's a single letter by itself.
  2. x+1 is an algebraic expression, not a variable.
  3. 1 is a number, not a variable.
  4. x+1=2 contains an equals sign (=) and is an equation, not a variable. You'll learn more about equations later.

So, only choice I represents a variable.

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Practice: Identifying Variables

Which of the following is a variable?

  1. $4$
  2. $a$
  3. $b+4$
a
 I only
b
 I, II, and III
c
 II only
d
 I and II
e
 III only
Practice: Identifying Variables

Which of the following is a variable?

  1. $h$
  2. $k-2$
  3. $3$
a
 III only
b
 I only
c
 I and II
d
 I, II, and III
e
 I and III
Example: Translating Between Phrases and Algebraic Expressions Containing Addition

Translate the following phrase into an algebraic expression: "The sum of three and a number."

EXPLANATION

If we denote our number by x , then "the sum of three and a number" can be written as 3+x.

Note: There are other correct answers as well. We could choose another letter, such as a , to represent the number. In that case, "the sum of three and a number" could also be written as 3+a.

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Practice: Translating Between Phrases and Algebraic Expressions Containing Addition

Translate the following phrase into an algebraic expression: "Eight more than a number."

a
 $\dfrac{x}{8}$
b
 $x+8$
c
 $x$
d
 $x-8$
e
 $8x$
Practice: Translating Between Phrases and Algebraic Expressions Containing Addition

Translate the following phrase into an algebraic expression: "Two more than a number."

a
 $2a$
b
 $a$
c
 $a+2$
d
 $\dfrac{a}{2}$
e
 $a-2$
Example: Translating Between Phrases and Algebraic Expressions Containing Subtraction

Translate the following phrase into an algebraic expression: "The difference of eight and a number."

EXPLANATION

If we denote our number by x , then "the difference of eight and a number" can be written as 8-x.

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Practice: Translating Between Phrases and Algebraic Expressions Containing Subtraction

Translate the following phrase into an algebraic expression: "$22$ less than a number."

a
 $\dfrac{q}{22}$
b
 $q+22$
c
 $22-q$
d
 $22q$
e
 $q-22$
Practice: Translating Between Phrases and Algebraic Expressions Containing Subtraction

Translate the following phrase into an algebraic expression: "Seven fewer than a number."

a
 $k-7$
b
 $\dfrac{k}{7}$
c
 $k+7$
d
 $7k$
e
 $k$
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