Problem 71 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The algebraic expression is \(8x + 48\).

Step 1: Understand the Phrase

The given phrase is 'Eight times the sum of a number and six'. This indicates we are dealing with two operations: a sum and a multiplication.

Step 2: Identify the Components

Identify the expressions for the operations involved: 'a number' corresponds to our variable, which is \(x\), and 'six' is a constant. The phrase 'the sum of a number and six' suggests we add these two components.

Step 3: Write the Sum

Write the algebraic expression for the sum: \(x + 6\). This represents the component 'the sum of a number and six'.

Step 4: Apply Multiplication

According to the phrase, we must take the result from the sum, \(x + 6\), and multiply it by eight. This gives us the expression: \(8(x + 6)\).

Step 5: Expand the Expression

Distribute the multiplication over the sum: \(8(x + 6) = 8x + 48\). This is achieved by multiplying each term inside the parentheses by eight.

Key Concepts

Algebraic Operations

Algebraic operations form the backbone of algebra and allow us to manipulate expressions so we can better understand relationships between variables and constants. In our exercise, we encounter two fundamental operations: addition and multiplication.

• Addition: This signifies combining two quantities. In the exercise, we sum the variable \(x\) and the constant \(6\) resulting in \(x + 6\).


• Multiplication: This operation enlarges the value of a quantity by a specified number of times. We multiply the previously found sum \((x + 6)\) by 8, which scales the entire expression up.

Understanding the order of operations (often remembered with the acronym PEMDAS) is crucial. In this case, multiplication follows the addition, which dictates how we handle the expression \(8(x + 6)\). Thus, in algebra, identifying and correctly applying operations is vital for forming correct expressions.

Simplifying Expressions

Simplifying expressions helps to present algebraic equations in their simplest and most understandable form. Simplification ensures clarity and reduces the complexity of algebraic expressions.

In our example, after writing the expression \(8(x + 6)\), we simplify it by expanding. Expansion involves applying the distributive property, which means we need to multiply each term inside the parentheses by the term outside. Thus:

• Multiply \(x\) by 8, resulting in \(8x\).


• Multiply 6 by 8, giving \(48\).

The simplified expression becomes \(8x + 48\). This process transforms a more complex form into a straightforward one, which helps when further operations or evaluations are needed. Simplification is a skill in algebra that allows us to see expressions clearer and solve them more efficiently.

Variable Representation

Variables are symbols, frequently letters like \(x\), that stand in for unknown or changing values. In algebra, variables allow us to construct expressions that can represent a myriad of problems or relationships in general terms.

In our given problem, the variable \(x\) stands for 'a number,' which can be any real number. By assigning \(x\) as a placeholder, we create a flexible expression \(8(x + 6)\) that can adapt to different values simply by substituting \(x\) with actual numbers.

Understanding the concept of variables is essential because they:

• Enable us to solve equations and find unknowns.


• Help in forming general formulas and functions applicable in numerous scenarios.


• Provide clarity and structure in mathematical expressions, making it easier to grasp complex ideas.

In short, variable representation is a powerful tool in translating real-world situations into algebraic expressions, which can then be manipulated and understood mathematically.