Problem 29 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. the sum of a number and 4

Step-by-Step Solution

Answer

The algebraic expression for the given English phrase is \(x + 4\).

Step 1: Identify the mathematical operations

The English phrase in question is 'the sum of a number and 4'. Here, 'the sum of' suggests we are dealing with an addition operation.

Step 2: Associate the variable

The problem states that \(x\) represents the number. So, wherever the question refers to 'the number', it can be replaced with \(x\).

Step 3: Form the algebraic expression

Putting together the operation and the variable, the algebraic translation of 'the sum of a number and 4' will be \(x + 4\).

Key Concepts

Mathematical Operations

Mathematical operations are fundamental processes used to manipulate numbers and mathematical expressions. They form the backbone of algebra and include several types. The primary operations are addition, subtraction, multiplication, and division. Each operation serves a specific purpose in creating and solving mathematical problems.

When translating English phrases into algebraic expressions, identifying the correct mathematical operation is key. In the phrase 'the sum of a number and 4', we recognize that 'the sum of' indicates the addition operation. It's essential to translate these verbal cues accurately to form the correct algebraic expression. Beginners are encouraged to familiarize themselves with common terms and their corresponding operations to develop confidence in translating phrases into mathematical language.

Addition Operation

The addition operation is one of the fundamental mathematical operations used to combine two or more quantities into a single total. It is denoted by the plus sign (+). This operation is commutative, meaning the order of the numbers doesn't affect the result. For example, both 3 + 5 and 5 + 3 equal 8.

In the context of our exercise, the addition operation is explicitly mentioned with the phrase 'the sum of'. This indicates that two values, 'a number' represented by a variable and 4, need to be added together. The meaning is straightforward: when you perform the addition of these two values, you obtain a new expression. Here, it results in the expression \(x + 4\). Understanding this operation's role is crucial for correctly forming algebraic expressions from English phrases.

Variable Representation

Variable representation is a powerful concept in algebra where letters or symbols stand in for numbers. This concept allows for the manipulation of expressions and solving of equations that vary or have unspecified quantities. In the context of algebraic expressions, variables enable us to represent unknown or variable quantities without specifying exact numerical values.

In the exercise, 'the number' is represented by the variable \(x\). This means whenever the phrase mentions 'the number,' we substitute this part with \(x\). Variables are particularly useful because they can assume any value, allowing for general solutions to a wide range of problems. They simplify complex expressions and make them universally applicable, in terms of calculation and formulation. Remembering this can be very helpful when learning to convert English phrases into algebraic terms.