Problem 7 - Algebra for College Students

Question

Translate each verbal phrase into \(a\) mathematical expression using \(x\) as the variable. $$ \text { Twice a number, decreased by } 13 $$

Step-by-Step Solution

Answer

\(2x - 13\)

Step 1: Identify key components

Determine the essential elements of the phrase. The phrase includes 'twice a number' and 'decreased by 13'.

Step 2: Represent 'twice a number'

The phrase 'twice a number' can be translated to mathematical terms as '2 times a number', or mathematically as \(2x\).

Step 3: Incorporate 'decreased by 13'

The phrase 'decreased by 13' implies a subtraction of 13 from the previously obtained expression. Therefore, we subtract 13 from \(2x\), resulting in the final expression \(2x - 13\).

Key Concepts

verbal phrases in algebra

Understanding verbal phrases is key to solving algebra problems. Verbal phrases describe a mathematical action in words.
For example, 'twice a number' or 'decreased by 13'. To solve these problems, recognize the operation described in the phrase.
Common phrases include:

• 'twice a number' (multiply by 2)
• 'sum of' (addition)
• 'difference' (subtraction)
• 'product' (multiplication)
• 'divided by' (division)

Understanding these basic phrases helps translate words into algebraic expressions.

mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operations. They form the building blocks of algebra.
In our example, we have 'twice a number', which converts to a mathematical term as '2 times x' or \(2x\).
An expression also involves certain operations like:

• Addition (+)
• Subtraction (-)
• Multiplication (x)
• Division (/)

Expressions do not have an equals sign (=), as they are not complete equations. Understanding expressions helps solve algebra problems effectively.

algebraic translation

Translating verbal phrases into algebraic expressions involves creating a mathematical representation of words.
Let's break it down step-by-step using our example.
First, 'twice a number' means we multiply a variable by 2. If we use x as the variable, it becomes \(2x\).
Next, 'decreased by 13' tells us to subtract 13. So, taking our earlier expression, we subtract 13: \(2x - 13\).
The final translation takes the form of an algebraic expression that accurately represents the given phrase. Practicing this skill helps to simplify and solve word problems in algebra.