Problem 11 - Algebra for College Students

Question

Translate each verbal phrase into \(a\) mathematical expression using \(x\) as the variable. The product of 8 and 16 less than a number.

Step-by-Step Solution

Answer

8 \( \times (x - 16)\)

Step 1: Identify the Variable

Let's choose a variable for the unknown number. Here, we use the variable 'x'. Further steps will involve forming the expression using this variable.

Step 2: Understand '16 less than a number'

The phrase '16 less than a number' means we start with the number (which is 'x') and then subtract 16: \(x - 16\).

Step 3: Find the Product

The problem states 'the product of 8 and 16 less than a number'. Therefore, we multiply 8 by the expression we formed in Step 2: 8 \( \times (x - 16)\).

Key Concepts

algebraic expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They are a way to represent real-world problems in a mathematical form. For example, instead of saying 'eight more than a number', we can write it as \( x + 8 \). This makes it easier to perform calculations and solve problems.
Algebraic expressions can include:

• Numbers (constants)
• Variables (like \( x \) or \( y \))
• Operators (like +, -, *, /)

Understanding how to translate verbal phrases into algebraic expressions is a crucial skill in algebra.

variable identification

A variable is a symbol used to represent an unknown value. In algebra, letters like \( x \), \( y \), or \( z \) are commonly used as variables. The main purpose of a variable is to act as a placeholder for an unknown quantity.
In our problem, the phrase '16 less than a number' indicates we need a variable to represent 'a number'. We choose \( x \) as our variable, making it easier to build our expression step by step.
Identifying variables helps to:

• Frame problems clearly
• Simplify complex expressions
• Enable solving of equations

Choosing the right variable and understanding its role is critical for translating verbal phrases into algebraic expressions.

product in algebra

In algebra, the term 'product' refers to the result of multiplying two or more numbers or expressions. It is one of the key operations in forming algebraic expressions.
To form the product of two numbers or expressions, use the multiplication operator (\( \times \)) or simply place the variables/constants next to each other (e.g., \( 8x \) instead of \( 8 \times x \)).
In our solution, the phrase 'the product of 8 and 16 less than a number' leads us to multiply 8 by the expression representing '16 less than a number'. This gives us \( 8 \times (x - 16) \), which simplifies to \( 8(x - 16) \).
Understanding the concept of the product is essential for simplifying and solving algebraic expressions.

subtraction in algebra

Subtraction in algebra is similar to basic arithmetic subtraction but often involves variables. To subtract one quantity from another, we use the minus sign (\( - \)).
In our exercise, the phrase '16 less than a number' translates to subtracting 16 from \( x \). Thus, we write it as \( x - 16 \).
It's important to understand:

• Which quantity is being subtracted from which
• The order of operations, as subtraction is performed after multiplication and division

In our example, recognizing '16 less than a number' correctly as \( x - 16 \) is a vital step. This ensures the correct formation and further simplification of the entire algebraic expression.