Problem 3 - Algebra 1: Concepts and Skills

Question

Match the phrase with its variable expression. Let \(x\) represent the number. A number increased by 11 A. \(x-11\) B \(\cdot x+11\) C. \(\frac{x}{11}\) D. \(11 x\)

Step-by-Step Solution

Answer

The correct match for the phrase 'A number increased by 11' is B. \(x+11\)

Step 1: Analyze the Phrase

First, identify the meaning of the phrase 'A number increased by 11'. Here, 'increased by' indicates addition.

Step 2: Match the Phrase to the Correct Expression

Look at each expression provided, A. \(x-11\) B. \(x+11\) C. \(\frac{x}{11}\) D. \(11 x\). It becomes evident that option B fits the explanation from step 1. In this case, 'x' is the number and it is 'increased by' or added to 11.

Key Concepts

Variable Expressions

When we talk about variable expressions in algebra, we're referring to combinations of numbers, variables (like x), and arithmetic operations (such as addition, subtraction, multiplication, and division) that stand for a specific value, depending on what value the variable takes.

For instance, in the exercise 'A number increased by 11', the variable x represents an unknown number. By combining x with the constant '11' through the operation of addition, we get the variable expression x + 11.

This basic form of algebraic expression is the foundation of more complicated problems in algebra. It's essential to understand how to correctly join numbers and variables using arithmetic operations to construct accurate expressions that match given phrases or real-world situations.

Translating Phrases to Algebra

The process of translating phrases into algebra involves converting words into symbols and numbers that can be resolved mathematically. It's like learning a new language where certain words correspond to specific algebraic operations.

Consider the following translation guidelines:

• Words like 'increased by', 'more than', or 'added to' usually indicate addition.
• 'Decreased by', 'less than', or 'subtracted from' suggest subtraction.
• 'Times', 'multiplied by', or simply the use of an adjective before a noun (for instance, 'twice a number') point to multiplication.
• Finally, 'divided by', 'split', or 'ratio of' refer to division.

In the given exercise, 'A number increased by 11' is accurately translated to the algebraic expression x + 11. This skill of translating is critical for understanding and solving algebra problems effectively.

Basic Algebra

Basic algebra is the segment of mathematics that deals with understanding how to manipulate algebraic expressions and equations, using various properties and operations. At its core, it involves working with variables, constants, and the relationships between them.

In essence, algebra teaches us not only to perform computational tasks, but to apply logical reasoning to find unknown values, or 'solve for x', so to speak. Basic algebraic principles include:

• The commutative property, where the order of addition or multiplication does not affect the result.
• The associative property, indicating how numbers are grouped in addition or multiplication can also change results.
• The distributive property, which allows for the multiplication across a sum or difference.

Understanding these properties helps one to manipulate and simplify expressions and is fundamental in working through algebra problems and scenarios.