Problem 74 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression and simplify if possible. Let \(x\) represent the unknown number. Half a number minus the product of the number and eight

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-15x}{2} \).

Step 1: Identify Each Part of the Phrase

The phrase 'half a number' can be represented as \( \frac{x}{2} \). The phrase 'the product of the number and eight' can be represented as \( 8x \).

Step 2: Formulate the Full Expression

According to the phrase, we need to subtract the product from half the number, so the expression becomes \( \frac{x}{2} - 8x \).

Step 3: Simplify the Expression

Combine the terms in the expression. First, convert \( \frac{x}{2} \) to the same form as \( 8x \) by expressing it as \( 0.5x \) or \( \frac{x}{2} = \frac{x}{2} \).

Step 4: Common Denominator

Express both terms with a common denominator: \( \frac{x}{2} - \frac{16x}{2} = \frac{x - 16x}{2} \).

Step 5: Simplify Further

Subtract within the numerator: \( \frac{x - 16x}{2} = \frac{-15x}{2} \).

Step 6: Conclusion: Simplified Expression

The simplified expression for the given phrase is \( \frac{-15x}{2} \).

Key Concepts

Simplification of Expressions

Simplifying algebraic expressions is a key skill in mathematics. It involves combining like terms and reducing expressions to their simplest form. The primary goal is to make expressions easier to understand and solve. For instance, when we have an expression such as \( \frac{x}{2} - 8x \), we're dealing with terms that can potentially be combined.

The expression \( \frac{x}{2} \) represents half of our unknown variable \( x \), while \( 8x \) represents 8 times the unknown variable. By expressing all components of an expression using a common base or denominator, we can effectively combine like terms.

• Convert terms to the same form by using a common denominator.
• Combine the terms by performing addition or subtraction as needed.

This process simplifies and reorganizes the expression, making it manageable and organized. It is a critical step before solving equations or performing further algebraic operations.

Product and Sum

The concepts of 'product' and 'sum' are essential foundations in arithmetic and algebra. They are the building blocks for creating algebraic expressions. Understanding these enables you to form equations and expressions correctly.

The product refers to the result you get when multiplying values together. In the expression \( 8x \), the 'product of the number and eight' means the unknown number \( x \) multiplied by eight. On the other hand, the sum is what results when you add two or more values together.

• The phrase "the sum of..." indicates an addition operation in algebraic translations.
• The term "the product of..." indicates multiplication.

Thus, tackling expressions often involves identifying these key operations—whether it’s combining for a sum, or multiplying for a product—and translating them into the corresponding algebraic terms.

Unknown Variable Manipulation

Manipulating unknown variables is a vital aspect of solving algebraic expressions. When working with variables, you're essentially making adjustments and rearrangements so that you can either simplify the expression or find specific values.

For example, in the expression \( \frac{x}{2} - 8x \), identifying that \( x \) is our variable allows us to conduct the necessary operations to consolidate terms. Here's how you can manipulate variables effectively:

• Represent unknown variables clearly (e.g., using \( x \) for simplicity).
• Use operations like addition, subtraction, multiplication, and division to combine or simplify terms.
• Identify opportunities to factor out common elements to reduce complexity.

This form of manipulation is foundational to solving equations, as it enables you to focus on re-organizing terms to arrive at a clearer, more workable form. By mastering this, you become better equipped to handle increasingly complex algebraic tasks.