**Algebraic Representation of Statements**

In order to solve word problems in algebra, it is necessary to translate the verbal statements into algebraic expressions. An unknown can be designated by a variable, usually x. For example, the statement "four times a number plus ten" can be written algebraically as "4x + 10."

**Examples**

**Example 1**

Write each of the following statements in symbols:

1. the sum of a number and 15

2. seven subtracted from two times a number

3. the product of six and a number

4. three times the sum of a number and 5

5. a number divided by four

**Solution 1**

1. x + 15

2. 2x – 7

3. 6x

4. 3(x + 5)

5. x ÷ 4

**Representing Variables in Word Problems with Two Quantities**

In solving word problems it is also necessary to represent two quantities using the same variable. For example, if the sum of two numbers is 10, and one number is x, the other number would be 10 – x. The reason is that given one number, say 7, you can find the other number by subtracting 10 – 7 to get 3. Another example: suppose you are given two numbers and the condition that one number is twice as large as the other. How would you represent the two numbers? The smaller number would be x, and since the second number is twice as large, it can be represented as 2x.

**Examples**

**Example 2**

Represent algebraically two numbers such that one number is three more than twice the other number.

**Solution 2**

Let x = one number

Let 2x + 3 = the other number

**Example 3**

Represent algebraically two numbers such that the difference between the two numbers is 7.

**Solution 3**

Let x = one number

Let x – 7 = the other number

**Algebraic Representation of Statements Practice Problems**

**Practice**

1. Represent algebraically two numbers so that one number is 5 more than another number.

2. Represent algebraically two numbers so that their sum is 50.

3. Represent algebraically two numbers such that one number is five times larger than another number.

4. Represent algebraically two numbers such that one number is half another number.

5. Represent algebraically two numbers such that one number is 10 more than twice the other number.

**Answers**

1. x, x + 5

2. x, 50 – x

3. x, 5x

4. x, x or x, 2x

5. x, 2x + 10

Practice problems for this concept can be found at: Expressions And Equations Practice Test.

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