Problem 124 - Algebra A Combined Function

Question

Translate each phrase to an expression. Use \(x\) to represent "a number." Multiply a number by -17 .

Step-by-Step Solution

Answer

-17x

Step 1: Identify the Unknown

The phrase mentions 'a number,' which we need to represent using a variable. According to the exercise, use \(x\) to represent 'a number.' Thus, \(x\) is our variable.

Step 2: Understand the Operation

The phrase asks us to 'multiply a number by -17.' Multiplication is an arithmetic operation where we take a number and increase it by adding it to itself a specified number of times or in this case, scaling by -17.

Step 3: Construct the Expression

Given 'a number' is represented by \(x\) and we need to multiply this by -17, the mathematical expression formed by this description is \(-17x\).

Key Concepts

Variable Representation

In algebra, a variable is a symbol that represents a value or number we do not know yet. In many cases, this is just any number, and we often use letters like \(x\), \(y\), or \(z\) to stand in place of these numbers. In this particular exercise, we are told to use \(x\) to represent "a number." This means that wherever we see \(x\), we know it signifies this unknown number.
Variables are crucial because they allow us to formulate expressions and equations that can describe real-world situations or general mathematical principles. For instance:

• If you buy \(x\) candies and each candy costs $2, then the total cost = \(2x\).
• In the problem given, \(x\) is simply standing in for the number we wish to multiply.

Arithmetic Operations

Arithmetic operations are fundamental mathematical processes like addition, subtraction, multiplication, and division. These operations help us manipulate numbers and quantities to solve problems or express mathematical ideas. In basic terms:

• **Addition** combines numbers to make a larger one.
• **Subtraction** finds the difference between numbers.
• **Multiplication** is repetitive addition or scaling a number.
• **Division** breaks a number into equal parts.

In the given exercise, we focus on multiplication, where we need to apply a scaling operation. The phrase "multiply a number by \(-17\)" involves taking the number we represent as \(x\) and scaling it by \(-17\). This operation balances between finding a product (result of multiplication) and understanding scaling in more complex expressions.

Multiplication in Algebra

Multiplication in algebra is slightly different from basic multiplication.While it still involves the same core idea of combining quantities, it often requires dealing with unknowns and variables like \(x\). When multiplying a variable by a constant (a fixed number), you're scaling that variable:

• **Example**: 'Multiply a number by \(-17\)' translates to \(-17x\), where \(-17\) is the scale factor for \(x\).

This means that whatever value \(x\) takes, \(-17x\) will always be \(-17\) times that value.
Such operations allow expressions to represent relationships and patterns in math and the real world, making it easier to model and solve problems. Remember:

• Keep track of negative signs, as in this case multiplying by \(-17\) reverses the sign of the number you are scaling.