Problem 13 - College Algebra

Question

Let \(x\) represent the number. Write each English phrase as an algebraic expression. Four times the sum of a number and 12

Step-by-Step Solution

Answer

The algebraic expression for the English phrase is 4*(x + 12).

Step 1: Identify Elements and Operations

The unknown number is represented by x. The phrase refers to performing 'four times' an operation, which signifies multiplication. The operation to be performed is the 'sum of a number and 12' which corresponds to addition.

Step 2: Translate the 'Sum' Operation

'Sum of a number and 12' can be represented as (x + 12), where x is the unknown number.

Step 3: Translate the 'Four Times' Operation

The 'four times' operation applied to the 'sum of a number and 12' would multiply the result of the sum by 4. The translation of the full phrase to an algebraic expression would then be 4*(x + 12).

Key Concepts

Translation of Phrases

When tackling algebraic expressions, understanding how to translate words into mathematical symbols is key. For example, the phrase "Four times the sum of a number and 12" requires breaking down each part into its mathematical equivalent.

• "Four times" indicates multiplication, suggesting the number four will multiply something.
• The "sum of a number and 12" tells us to add a number, usually represented by the variable \(x\), to twelve.

Combining these translations, we convert the entire sentence into an appropriate algebraic expression. Breaking it down step by step helps simplify and clarify the meaning. In this case, the final expression becomes \(4 \times (x + 12)\). By dissecting word problems systematically, you learn to bridge the gap between language and algebra.

Multiplication and Addition

Multiplication and addition are fundamental operations in algebra that frequently appear together in problems.

• "Addition" combines numbers to get their sum, as seen with "a number and 12" becoming \(x + 12\).
• "Multiplication" then takes the entire result of a sum and scales it, represented by "four times" which translates to \(4 \times (x + 12)\).

Understanding these operations is crucial because they often work in unison in expressions. As in the given phrase, multiplying after completing an addition ensures proper order of operations. It's common practice to use parentheses to clarify operations and maintain the correct sequence, such as \( (x + 12) \) being multiplied by four. Learning to correctly perform these operations helps in accurately solving or simplifying algebraic expressions.

Unknown Variables

Variables are symbols used to represent unknown values in mathematical expressions or equations. In our example, the variable \(x\) symbolizes the unknown number in the expression \(4 \times (x + 12)\).

• Variables serve as placeholders that can change or adapt as needed within problems.
• They allow flexibility in writing and solving equations, permitting a universal approach to different scenarios.

When dealing with variables, recognize that they can stand for any number and understanding how to manipulate them is key to solving equations. In algebra, you will often assign a variable to represent quantities in word problems to form a working equation. This technique simplifies both the setting up and solving of complex problems. Ultimately, mastering the use of variables is an essential skill for progressing in algebra.