Problem 11 - Elementary Algebra Within Reach

Question

In Exercises \(11-22\), translate the verbal phrase into an algebraic expression. $$ x \text { increased by } 5 $$

Step-by-Step Solution

Answer

The algebraic expression for 'x increased by 5' is \(x + 5\)

Step 1: Identify the Variable

The variable in the expression has been denoted as 'x'.

Step 2: Translate Verbal Phrase

The phrase 'increased by' indicates addition in algebra. So, 'x increased by 5' translates to 'x plus 5'.

Key Concepts

Translating Verbal Phrases

Translating verbal phrases into algebraic expressions is a crucial skill in algebra. It involves converting words into mathematical symbols and operations. This process requires a clear understanding of the language used in mathematical contexts.

Key Steps:

• Identify Keywords: Recognize words that indicate mathematical operations. For example, 'increased by' signals that addition is to be used.
• Determine the Variable: Understand what variable or expression represents the unspecified quantity. In our example, it is denoted by 'x'.
• Translate: Replace the words with appropriate mathematical symbols. As seen, 'x increased by 5' translates into 'x + 5'.

The aim is to make a seamless connection between verbal and algebraic language, allowing equations to be formed and manipulated.

Basic Algebra Concepts

Algebra forms one of the foundational areas of mathematics, centered around using symbols to represent numbers and operations. These symbols are the building blocks in creating equations and expressions.

Important Concepts:

• Variables: Letters like 'x' or 'y' that stand for unknown values. These symbols can vary, hence the name 'variable'.
• Expressions: Combinations of variables, numbers, and operations (such as addition, subtraction, etc.) without an equality sign. For instance, 'x + 5'.
• Operations: Basic arithmetic operations form the backbone of algebraic manipulation, such as addition, subtraction, multiplication, and division.

Learning basic algebra concepts helps in problem-solving and understanding more advanced mathematical ideas. By mastering these basics, complex problems become easier to tackle.

Addition in Algebra

Addition in algebra operates under the same principles as basic arithmetic but is applied to variables and expressions. It involves combining numbers and/or variables to form new expressions.

How It Works:

• Combining Like Terms: You can add together terms that have the same variables and exponents. For example, '2x + 3x' simplifies to '5x'.
• Adding Constants: Just as in regular arithmetic, you can add numerical constants straightforwardly, like '3 + 5 = 8'.
• Creating New Expressions: When adding variables with constants, such as in 'x + 5', you form a new expression indicating 'x' is increased by 5.

Understanding addition in algebra is vital for creating and simplifying expressions, solving equations, and tackling complex problem-solving scenarios.