Problem 21 - Algebra 1: Concepts and Skills

Question

Match the sentence with its equation. Let x represent the number. The product of 2 and a number is 4. A. \(x-4=2\) B. \(x+2=4\) C. \(\frac{x}{4}=2\) D. \(2 x=4\)

Step-by-Step Solution

Answer

The sentence 'The product of 2 and a number is 4' matches with the equation D: \(2x = 4\).

Step 1: Understand Sentence Terminology

First, it's important to understand the terminology used in the sentence. 'Product' means multiplication, 'is' means equals, and 'a number' refers to x.

Step 2: Match Sentences to Equations

The operation is multiplication (product) and the number is 2. The number it equals to is 4. So, the product of 2 and a number (x) must be an equivalent of 4. When we check all the equations, we can see that the sentence matches equation D: \(2x = 4\). None of the others includes a multiplication of 2 and x which equals 4.

Step 3: Confirm Correct Answer

We now need to confirm that \(2x=4\) is the correct answer. The only solution for x is 2 because \(2 * 2 = 4\). Thus, verifying that equation D is correct.

Key Concepts

Understanding Algebraic Terminology

When embarking on the journey of algebra, we often encounter a language of its own. Algebraic terminology is the vocabulary that turns words into symbols, setting the stage for problem-solving. For instance, take the phrase 'the product of 2 and a number.' To proceed, we need to decode the word 'product.' In algebra, 'product' signifies multiplication. The 'number' in question is typically represented by a variable, like 'x', which serves as a placeholder until its value is identified. The word 'is' in algebra functions as an equal sign ((=)), demonstrating the relationship between both sides of an equation, pointing us towards balance.
For students, understanding this terminology is like learning a new language, one that translates verbal expressions into mathematical ones, allowing for clear and concise communication of quantitative ideas.

Identifying Mathematical Operations

In our example, 'the product of 2 and a number is 4,' we are tasked to single out the appropriate mathematical operation. Identifying and understanding basic operations like addition, subtraction, multiplication, and division is fundamental. They are the building blocks of more complex equations and are signified by specific terms:

• Addition is noted by 'sum,' 'plus,' or 'increased by.'
• Subtraction goes by 'difference,' 'minus,' or 'decreased by.'
• Multiplication is referred to as 'product,' 'times,' or 'of.'
• Division is indicated by 'quotient,' 'divided by,' or 'per.'

Once we've identified the operation— in this case, multiplication (the product)—we can translate the phrase into an equation precisely, ensuring each part aligns with its mathematical counterpart.

Solving Simple Equations

With the algebraic expression in hand, the next step is to solve it. Solving simple equations begins with isolation of the variable we're solving for, which in this exercise is 'x.' In our equation (2x = 4), we aim to find the value of 'x' that makes the statement true. To do this, we perform the inverse operation. Since 'x' is being multiplied by 2, we divide both sides of the equation by 2, obtaining (x = 2). This result is the essence of an algebraic solution—it's precise, unambiguous, and checkable. It provides a sure foundation on which to build more advanced mathematical skills.
A frequent advice for students is to always verify their answers. By substituting 'x' back into the original equation, we confirm if the solution is consistent. This step is not only good practice but a method to understand the importance of accuracy within algebra.