Problem 112 - Introductory and Intermediate Algebra for College Students 4th

Question

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The product of \(\frac{3}{4}\) and a number increased by 9

Step-by-Step Solution

Answer

The algebraic expression that represents the English phrase is \(\frac{3}{4}x + 9\).

Step 1 - Identify the Unknown

The exercise mentions a number, but does not give a specific value. This is our unknown and we'll represent it using the variable \(x\).

Step 2 - Begin Translation

The first part of the sentence, 'The product of \(\frac{3}{4}\) and a number', translates into an operation of multiplication between \(\frac{3}{4}\) and our unknown number \(x\). Therefore, the initial translation is \(\frac{3}{4}x\).

Step 3 - Complete Translation

The second part of the sentence, 'increased by 9', indicates an operation of addition of 9 to the result of the previous operation. So, we add 9 to our existing translation '\(\frac{3}{4}x\)', leading to the completed expression: \(\frac{3}{4}x + 9\).

Key Concepts

Mathematical Translation

When we talk about mathematical translation, it refers to the process of converting words from a problem statement into a mathematical expression or equation. This is an essential skill in algebra that helps us in understanding and solving problems effectively. It is much like translating one language into another. For example, consider the phrase "the product of \(\frac{3}{4}\) and a number." Here, "the product" suggests multiplication. Hence, you multiply \(\frac{3}{4}\) by the unknown number, represented as \(x\). Therefore, this part of the phrase translates to \(\frac{3}{4}x\).After understanding this portion, we look at "increased by 9," which tells us to add 9 to our previous result. Thus, the original statement converts into the expression \(\frac{3}{4}x + 9\). Translating step by step helps us maintain accuracy and ensure we account for each part of the given information.

Algebraic Operations

Algebraic operations are the basic math processes used within algebra, which usually include addition, subtraction, multiplication, and division. Understanding these operations is crucial for forming and solving algebraic expressions.In this context, two main operations are involved: multiplication and addition.

• Multiplication: In the phrase "the product of \(\frac{3}{4}\) and a number," multiplication takes precedence. This operation combines \(\frac{3}{4}\) with the variable \(x\), resulting in \(\frac{3}{4}x\).
• Addition: The latter operation, "increased by 9," signals that we need to add 9 to the previously obtained result, yielding \(\frac{3}{4}x + 9\).

By understanding how and when to apply each operation, converting a word problem into a mathematical expression becomes clearer and more manageable.

Unknown Variables

In algebra, unknowns are typically represented by variables, the most common being \(x\), \(y\), or \(z\). These variables act as placeholders for numbers we do not know yet. In our exercise, the phrase "a number" indicates that there is an unknown quantity we must represent. Here, we use \(x\) to stand for this unknown number.Recognizing unknowns is foundational in creating expressions and equations. By assigning them a variable:

• You provide a symbol that can interact with constants and other variables through algebraic operations.
• Allows for manipulation and problem-solving using algebraic rules.

The process of using variables not only makes solving equations possible but also enhances our understanding of relationships between different components of a problem.