Problem 72 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression and simplify if possible. Let \(x\) represent the unknown number. Six times the difference of a number and five

Step-by-Step Solution

Answer

The expression is \(6x - 30\).

Step 1: Identify Key Phrases

The phrase we have is 'Six times the difference of a number and five.' The important parts to identify are the operations and the unknown variable, which is given as \(x\).

Step 2: Translate Phrases into Expression

The word 'difference' implies subtraction. 'The difference of a number and five' translates to \(x - 5\). The phrase 'six times' implies multiplication, so six times the expression \(x - 5\) is written as \(6(x - 5)\).

Step 3: Simplify the Expression

The expression \(6(x - 5)\) can be simplified by distributing the 6 to each term inside the parentheses. Thus, \(6 \cdot x - 6 \cdot 5 = 6x - 30\).

Key Concepts

Simplifying Expressions

Simplifying an algebraic expression involves reducing it to its most concise form. This can often mean performing operations like addition, subtraction, multiplication, or division to combine like terms or eliminate unnecessary elements. When we simplify, we're aimed at making the expression easier to understand or work with. In our exercise, we start with the expression \(6(x - 5)\). By applying simplification techniques, particularly the distributive property (which we'll explore further in another section), we break down the expression step-by-step.Here's how we can approach it:

• Original Expression: \(6(x - 5)\)
• Apply Distributive Property: \(6 \cdot x - 6 \cdot 5\)
• Final Simplified Expression: \(6x - 30\)

Notice how the expression is now a simple linear equation: \(6x - 30\), which is much easier to understand and use in further calculations or solving.

Translating Phrases to Algebra

Translating phrases into algebraic expressions is a key skill in algebra that allows us to move between verbal statements and mathematical ones. This involves understanding the words and identifying common mathematical operations they imply. Consider the phrase from our example, 'Six times the difference of a number and five'. Here, it's crucial to:

• Recognize 'difference' as subtraction, leading to the expression \(x - 5\).
• Understand 'six times' as a multiplier (multiplication), so six times the difference becomes \(6(x - 5)\).

These translations help frame mathematical problems in a familiar and logical way, improving our ability to solve them and interpret solutions correctly. It's like decoding a language where each word has a corresponding mathematical symbol or operation.

Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions and making calculations easier. This property states that a single term multiplied by a sum or difference inside a parenthesis can be distributed to each of the terms inside. For example, in the expression \(6(x - 5)\), you use the distributive property by multiplying 6 by each term inside the parentheses:

• Multiply 6 by \(x\) to get \(6x\).
• Multiply 6 by \(-5\) to get \(-30\).

So, \(6(x - 5)\) simplifies to \(6x - 30\). This process helps us break down complex expressions into simpler ones, making them more manageable for further mathematical operations. The distributive property is especially handy when dealing with expressions that contain multiple terms or factors.