Problem 5 - Algebra 1: Concepts and Skills

Question

Match the phrase with its variable expression. Let x represent the number. The difference of a number and 11 A. \(x-11\) B \(\cdot x+11\) C. \(\frac{x}{11}\) D. \(11 x\)

Step-by-Step Solution

Answer

The phrase 'The difference of a number and 11' corresponds to the expression \(x-11\). Therefore, answer A is correct.

Step 1: Identifying the Operation

In the phrase 'The difference of a number and 11', the keyword to notice is 'difference'. In mathematics, the word 'difference' indicates the operation of subtraction.

Step 2: Matching the Phrase to the Expression

Now, we need to match the phrase to a potential expression that correctly shows the operation of subtraction. This matching process involves comparing the phrase to each listed expression until finding the one that suits. In this case, we're looking for a subtraction of 11 from a variable 'x'.

Step 3: Finding the Correct Expression

We match the phrase to expression A. \(x-11\). This expression correctly represents the subtraction of 11 from a number 'x'.

Key Concepts

Difference in Mathematics

The term "difference" is fundamental in mathematics, especially when dealing with arithmetic operations. It's important to understand that when we talk about the "difference" between two numbers, we are referring to the outcome of a subtraction operation.
This operation takes two operands, where one number is subtracted from another.
In the context of variable expressions, the difference usually means we are removing a specific quantity from a variable. When we say "the difference of a number and 11," we specifically look for an expression that shows one number being taken away from another.

• The first number, often represented by a variable like 'x', is the minuend.
• The second number, in this case, 11, is the subtrahend.
• The result is called the difference.

Understanding the terminology and setup is crucial in identifying and writing correct variable expressions. By knowing that a "difference" signifies subtraction, we can properly match phrases to algebraic expressions.

Subtraction Operation

Subtraction is one of the four basic arithmetic operations, alongside addition, multiplication, and division. It involves taking away one number from another.
In a subtraction operation, the components consist of the minuend, the subtrahend, and the difference.
When working with variable expressions, the goal is to denote this arithmetic relationship using symbols and letters.

• The minuend is what you start with; in algebraic terms, this can be a variable such as 'x'.
• The subtrahend is the amount you subtract, typically a constant or another variable.
• The result, or the difference, is what's left after the subtraction is carried out.

For example, if the "difference of a number and 11" is given, this translates to the subtraction operation: a variable 'x' minus 11.
Written mathematically, this is expressed as: \(x - 11\).
Understanding subtraction in terms of variables is vital as it opens the door to solving more complex algebraic equations and problems.

Algebraic Expressions

Algebraic expressions are mathematical expressions that use numbers, variables, and operations to convey a specific relationship or formula. They are the foundation of algebra and play a critical role in expressing mathematical ideas clearly and succinctly.
Variables in these expressions represent unknown values and are often denoted by letters such as x, y, or z.
An expression can include:

• Constants, which are fixed values like 11, 4, or \pi.
• Variables, which stand in for unknown or changing quantities.
• Operations such as addition, subtraction, multiplication, and division.

The challenge in writing algebraic expressions is translating verbal descriptions into mathematical notation.
For example, "the difference of a number and 11" becomes the expression \(x - 11\), where:

• 'x' is the variable representing the number.
• '-11' indicates subtraction of 11 from the variable.

Mastering the creation of algebraic expressions allows for successful problem-solving across many fields of study, making it an essential skill in mathematics.