Problem 85 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression. Let \(x\) represent the unknown number. Three times a number, increased by 22

Step-by-Step Solution

Answer

The expression is \(3x + 22\).

Step 1: Identify the Unknown

First, identify the unknown quantity in the problem. In this case, it is mentioned as 'a number,' which we will represent using the variable \(x\).

Step 2: Translate Verbal Phrases to Algebraic Expressions

The phrase 'three times a number' translates mathematically to \(3x\), because you multiply the unknown number \(x\) by 3.

Step 3: Include the Increase

The next part of the phrase is 'increased by 22.' This means you add 22 to the previous expression. So, you would write this as \(3x + 22\).

Step 4: Construct the Expression

Combine the translated parts of the phrase to form the complete algebraic expression. So, the phrase 'three times a number, increased by 22' becomes \(3x + 22\).

Key Concepts

Variable Representation

In algebra, representing unknown quantities is fundamental, and we do this with variables. A variable is essentially a symbol, often a letter like \( x \), that stands in for a number we don’t yet know. This representation allows us to work with the unknown mathematically until we have enough information to solve for its value. For example, in the exercise, we use \( x \) to represent "a number." By using \( x \), we create a placeholder for whatever value the unknown will eventually be, enabling us to easily manipulate and solve expressions and equations. Variables bring flexibility and precision to algebra, making them a core component of mathematical problem-solving and representation.

Translating Verbal Phrases

Translating words into mathematical language is a key skill in algebra. This helps us convert everyday language into expressions that can be manipulated mathematically. For instance, in the phrase "three times a number," we recognize the phrase "a number" as our variable \( x \). The word "times" suggests multiplication, so "three times a number" translates to \( 3x \).

Understanding common vocabulary is essential here:

• 'Sum' indicates addition.
• 'Product' denotes multiplication.
• 'Difference' points to subtraction.
• 'Quotient' implies division.

Recognizing these terms allows us to construct precise mathematical statements from verbal descriptions. Effective translation bridges the gap between verbal problems and mathematical solutions.

Constructing Expressions

Once you've translated the individual verbal components into mathematical terms, the next step is to construct an algebraic expression. This involves combining the translated parts with proper mathematical operations to express the complete idea. For our specific example, we start with "three times a number," written as \( 3x \), and then we identify that it needs to be "increased by 22."

This requires us to add 22 to the result, resulting in the expression \( 3x + 22 \).

• Identify each part of the phrase.
• Translate each element to a mathematical term.
• Combine these into a singular algebraic expression.

This process is essentially about piecing together the translated parts to form a comprehensive mathematical representation, ensuring we accurately capture the essence of the verbal phrase.