Problem 18 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. Negative seven added to some number.

Step-by-Step Solution

Answer

The expression is \(x - 7\).

Step 1: Identify Variables

The phrase mentions 'some number,' which means we need to introduce a variable to represent it. Let's use \(x\) as the variable that represents this unknown number.

Step 2: Understand Numerical Values and Operations

The phrase 'Negative seven added to' implies a mathematical operation. 'Negative seven' is represented as \(-7\). The word 'added' indicates addition. Therefore, we need to add \(-7\) to our variable \(x\).

Step 3: Construct Mathematical Expression

Combine the information from the previous steps into a mathematical expression. Since we are adding \(-7\) to \(x\), the expression becomes \(x + (-7)\), which simplifies to \(x - 7\).

Key Concepts

Variables in Algebra

In algebra, variables play a crucial role as they stand in place of unknown numbers. These are typically represented by letters, like \(x\), \(y\), or \(z\). Introducing a variable allows us to construct expressions or equations when the actual value is not known yet.

Imagine you stumble upon a statement that refers to an unknown quantity, such as 'some number.' To make it concrete for mathematical operations, we introduce a variable, say \(x\), to represent this mystery number.

Using variables effectively opens the door to solving equations, exploring patterns, and performing general mathematical reasoning. The choice of variable isn’t strict—you can choose any letter—but it's often chosen to be intuitive or follow convention, like \(n\) for numbers.

Here’s the key takeaway: variables are placeholders that help us translate words into numbers and operations, giving us the tools to solve complex problems step-by-step.

Mathematical Operations

Operations are the building blocks of math that tell us what to do with numbers and expressions. In the problem at hand, we deal with addition, identified by the word 'added.' In everyday language, operational words signal these actions, translating verbal expressions into algebraic expressions.

Let's explore some basic operations:

• **Addition** is often indicated by words like 'sum,' 'added to,' or 'plus.'
• **Subtraction** uses cues such as 'less than,' 'minus,' or 'decreased by.'
• **Multiplication** is hinted at by terms like 'product,' 'times,' or 'multiplied by.'
• **Division** surfaces with words like 'quotient,' 'divided by,' or 'per.'

In the original exercise, you see the term 'negative seven added to,' which guides us to use the addition operation for combining \(-7\) and the variable \(x\). The language of math blends seamlessly from words to symbols, creating an expression or an equation you can solve further.

Recognizing these cues and executing the correct operation is essential in translating everyday language into the math we understand and work with.

Constructing Expressions

Constructing mathematical expressions from verbal phrases requires careful parsing of the sentence to identify variables and operations, followed by combining these parts correctly. Let's break it down with the exercise example:

The sentence 'Negative seven added to some number' gives us a clear path to follow. We identified our variable, \(x\), to denote 'some number,' and we noted '-7' as the value being added. Now it's about putting them together into an expression: \(x + (-7)\). This step involves more than arithmetic; it's about understanding and logically organizing mathematical concepts.

Once the expression \(x + (-7)\) is formed, you might recognize that it can be simplified to \(x - 7\), because adding a negative number is the same as subtracting.

Simplification is an important process, ensuring expressions are as straightforward as possible but still convey the same meaning. By following these steps, you practice translating words into math, a skill that helps in problem-solving and applying math to real-world situations.