Problem 61 - Prealgebra

Question

Write the mathematical expressions that are equivalent to each of the following English phrases. Twice the sum of a number and 6

Step-by-Step Solution

Answer

The expression is \( 2(x + 6) \).

Step 1: Understanding the Phrase

The phrase "twice the sum of a number and 6" requires us to translate each part into mathematical terms. "A number" represents an unknown value, which we can denote as a variable, typically 'x'. "Sum of a number and 6" implies we need to add the number and 6 together.

Step 2: Expressing the Sum

Translate the words "sum of a number and 6" into a mathematical expression. This gives us the expression \( x + 6 \), where 'x' represents the unknown number.

Step 3: Applying 'Twice' to the Sum

Now, we need to incorporate the "twice" part of the phrase. "Twice" means to multiply by 2. Therefore, we take our sum from the previous step and multiply it by 2, resulting in the expression \( 2(x + 6) \).

Step 4: Final Expression

The complete mathematical expression that represents the English phrase "twice the sum of a number and 6" is \( 2(x + 6) \). This expression correctly follows the operation order stated in the phrase.

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental components of algebra. They consist of numbers, variables, and operation symbols that collectively represent a value or a relationship. In this exercise, the expression "twice the sum of a number and 6" becomes the focal point. We translate this English phrase into the algebraic expression \( 2(x + 6) \).

• "Twice" implies multiplication by 2.
• "The sum of a number and 6" points to the addition operation, resulting in \( x + 6 \).

These components combine to form the entire algebraic expression: \( 2(x + 6) \). Algebraic expressions allow us to convey complex ideas succinctly using symbolic language. Every part of the expression carries meaning, enabling us to perform mathematical operations or solve equations effectively.

Variables in Algebra

In algebra, variables are symbols that stand in for unknown or changeable values. Typically represented by letters like 'x', 'y', or 'z', they serve as placeholders, making it possible to form expressions and equations.

• In our exercise, 'x' represents an unknown number.
• The variable helps create a dynamic expression that can be adjusted or solved based on additional information or conditions.

Variables are crucial because they allow for the abstraction of numbers, facilitating general solutions and understanding. By using 'x' in \( x + 6 \), we can explore different values of 'x' and determine how changes affect the overall expression. This abstraction is a key strength of algebra.

Order of Operations in Algebra

The order of operations in algebra ensures that all mathematicians solve expressions the same way. This consistency is crucial for accurately translating phrases into mathematical expressions, like in our exercise with \( 2(x + 6) \). Algebra follows specific rules often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Here's how we applied the order of operations to understand the phrase:

• Parentheses: First, we handle operations inside the parentheses \(x + 6\), representing "a number and 6."
• Multiplication: Next, we apply "twice" by multiplying the sum by 2.

Applying these systematic steps ensures precision and clarity, preventing any misinterpretation of the expression. By sticking to the order of operations, the expression \( 2(x + 6) \) exactly mirrors the given phrase "twice the sum of a number and 6."