Problem 80 - Beginning and Intermediate Algebra

Question

Write each phrase as an algebraic expression. Let \(x\) represent the unknown number. The product of 8 and a number, decreased by 10

Step-by-Step Solution

Answer

The algebraic expression is \(8x - 10\).

Step 1: Identify the Components

First, let's identify the components of the phrase. It mentions the 'product of 8 and a number' and 'decreased by 10'. So, the two main actions are multiplication and subtraction.

Step 2: Assign Variables

We are told to use the variable \(x\) to represent the unknown number. This will be the number that 8 is multiplied by.

Step 3: Formulate the Expression for the Product

The phrase 'the product of 8 and a number' can be expressed as \(8x\), since 'product' means multiplication.

Step 4: Apply the Decrease

Next, apply the phrase 'decreased by 10' to the expression \(8x\). This means we need to subtract 10, forming the equation \(8x - 10\).

Key Concepts

Product in Algebra

In algebra, understanding the concept of a "product" is essential. The term "product" refers to the result of multiplying two or more numbers. In this context, we are dealing with a phrase that includes the product of 8 and a variable, represented by an unknown number. It's important to note that in algebra, the "product" not only applies to numbers you already know but also to variables or unknowns. This means multiplying a known numeric value, such as 8, with a variable, such as the letter x, to form an expression like \(8x\). Recognizing that "product" implies multiplication is a basic building block in forming and understanding algebraic expressions.

Variable Representation

A vital part of algebra is using variables to stand in for unknown quantities. In our exercise, the variable \(x\) has been chosen to represent an unknown number. This concept of variable representation helps simplify complex mathematical ideas and allows for the formulation of general rules and equations. Variables can be letters, like \(x\), \(y\), or any other symbol, and they are placeholders for numbers you may not know yet. Understanding that \(x\) represents an unknown enables you to manipulate it algebraically, such as by multiplying it with other numbers, as we did with creating the expression \(8x\). The flexibility of variables is what makes algebra so powerful.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations form the foundation for all algebraic expressions. In the exercise, we focus on multiplication and subtraction. Multiplication comes up when forming the product of a known number, like 8, and a variable \(x\), leading to the expression \(8x\). After obtaining the product, subtraction is used to adjust the expression as specified by the problem. The subtraction "decreased by 10" modifies the expression \(8x\) to become \(8x - 10\). Recognizing and applying these operations are crucial skills in solving algebraic problems and writing expressions correctly.

Step-by-Step Solutions

Breaking down a problem into step-by-step solutions can hugely aid understanding. Here, each step builds on the previous one, providing clarity to achieve the solution. First, identify what the problem is asking—locating key math terms like "product" and "decreased by." Next, assign a variable to represent any unknowns; in this case, we use \(x\).

• Formulate the initial algebraic expression by identifying the multiplication involved, resulting in \(8x\).
• Then, apply arithmetic operations like subtraction to complete the expression as \(8x - 10\).

This structured method ensures that all aspects of the problem are considered, understood, and correctly communicated in a logical sequence. Step-by-step solutions are an excellent way to ensure no detail is overlooked and to develop a firm grasp on how to effectively solve mathematical problems.