Expression For Decreased Card Collection: X - 7

In the realm of mathematics, translating verbal phrases into algebraic expressions is a fundamental skill. It allows us to represent real-world scenarios using symbolic language, making it easier to solve problems and understand relationships between quantities. This article delves into the process of translating the phrase "Lamar decreased his collection of cards by seven" into an algebraic expression, exploring the concepts of variables, constants, and mathematical operations involved. We will dissect the phrase, identify its key components, and then construct the corresponding expression. Furthermore, we will analyze why certain expressions are correct while others are not, reinforcing the importance of precise language and mathematical notation. By the end of this discussion, you will have a solid understanding of how to represent subtraction in algebraic expressions and be well-equipped to tackle similar translation problems.

Decoding the Phrase: "Lamar decreased his collection of cards by seven"

To effectively translate the phrase "Lamar decreased his collection of cards by seven" into an algebraic expression, we must first break down the sentence and identify its essential components. The phrase revolves around the idea of a change in the size of Lamar's card collection. The key action verb here is "decreased," which indicates a reduction or subtraction. We need to represent the initial size of Lamar's collection and the amount by which it was decreased. Let's dissect the phrase step by step to understand its mathematical implications. The starting point is "Lamar's collection of cards." Since we don't know the exact number of cards Lamar initially had, we need to represent this unknown quantity with a variable. In algebra, a variable is a symbol, often a letter, that stands for a value that can vary. A common choice for a variable is "x," but we could use any letter. So, let's let "x" represent the number of cards Lamar had initially. The next crucial part of the phrase is "decreased by seven." The word "decreased" clearly indicates a subtraction operation. The phrase "by seven" tells us the specific amount that was subtracted. Thus, we need to subtract 7 from the initial number of cards, which we represented as "x." Now, we can combine these components to form the algebraic expression. We start with the initial quantity, "x," and then subtract 7 from it. This gives us the expression x - 7. This expression accurately represents the phrase "Lamar decreased his collection of cards by seven" because it shows the initial number of cards ("x") being reduced by 7. Understanding this breakdown is crucial for correctly translating verbal phrases into algebraic expressions. The ability to identify the variable, the operation, and the constant allows us to construct a mathematical representation that accurately reflects the given scenario. In the following sections, we will compare this correct expression with other options to highlight the importance of precision in mathematical notation.

Understanding the Correct Expression: x - 7

The algebraic expression x - 7 accurately represents the phrase "Lamar decreased his collection of cards by seven." To fully grasp why this is the correct representation, let's delve deeper into the meaning of variables, constants, and the subtraction operation within this context. The variable "x" in this expression symbolizes the initial number of cards in Lamar's collection. It's crucial to recognize that "x" can represent any non-negative integer since Lamar cannot have a negative number of cards. The constant "7" represents the specific quantity by which Lamar's collection was decreased. This is a fixed value, unlike the variable "x," which can change. The subtraction operation, denoted by the minus sign (-), is the core of this expression. It signifies that we are taking away 7 from the initial number of cards. The order of the terms in the expression is critical. We start with "x" because this is the initial quantity, and then we subtract 7 because Lamar decreased his collection. This order reflects the sequential nature of the phrase – Lamar had some cards, and then he reduced that number by seven. To illustrate further, let's consider a few examples. If Lamar initially had 15 cards, then x = 15, and the expression becomes 15 - 7, which equals 8. This means Lamar would have 8 cards left after decreasing his collection. If Lamar had 20 cards, then x = 20, and the expression becomes 20 - 7, which equals 13. Again, this shows the number of cards remaining after the decrease. These examples demonstrate how the expression x - 7 consistently and correctly represents the scenario described in the phrase. The variable "x" holds the initial quantity, and the subtraction of 7 accurately reflects the decrease in Lamar's card collection. Understanding this fundamental concept is essential for successfully translating verbal phrases into algebraic expressions. In the subsequent sections, we will contrast this correct expression with other options to highlight common errors and reinforce the importance of precise mathematical representation.

Analyzing Incorrect Options: Why 7 - x, x + 7, and 7x Are Wrong

While x - 7 correctly represents "Lamar decreased his collection of cards by seven," it's equally important to understand why other expressions are incorrect. Let's examine the options 7 - x, x + 7, and 7x to highlight the errors in their representation of the given phrase. First, consider the expression 7 - x. This expression represents subtracting the initial number of cards (x) from 7. This is the reverse of what the phrase describes. The phrase states that Lamar decreased his collection, meaning we start with the collection size and subtract from it, not the other way around. For example, if Lamar had 10 cards (x = 10), then 7 - x would be 7 - 10, which equals -3. It's impossible for Lamar to have a negative number of cards in his collection, so this expression doesn't make sense in the context of the problem. Therefore, 7 - x is an incorrect representation because it reverses the order of subtraction and doesn't align with the phrase's meaning. Next, let's analyze the expression x + 7. This expression represents adding 7 to the initial number of cards. The phrase, however, indicates a decrease, which implies subtraction, not addition. Adding 7 would mean Lamar increased his collection by seven cards, which contradicts the original statement. If Lamar had 10 cards (x = 10), then x + 7 would be 10 + 7, which equals 17. This means Lamar would have 17 cards, an increase rather than a decrease. Thus, x + 7 is incorrect because it uses addition instead of the required subtraction operation. Finally, let's consider the expression 7x. This expression represents 7 multiplied by the initial number of cards. Multiplication doesn't fit the scenario described in the phrase. Multiplication implies scaling or repeated addition, not a decrease or reduction. The phrase speaks of reducing the collection by a fixed amount (7), not multiplying it. If Lamar had 10 cards (x = 10), then 7x would be 7 * 10, which equals 70. This would mean Lamar has 70 cards, which is unrelated to the phrase's intended meaning of decreasing the collection. Therefore, 7x is incorrect because it uses multiplication, an operation that doesn't reflect the concept of decreasing the collection. In summary, each of these expressions – 7 - x, x + 7, and 7x – fails to accurately represent the phrase "Lamar decreased his collection of cards by seven" due to their incorrect use of mathematical operations and the order of terms. Understanding these errors helps reinforce the importance of precise translation and accurate representation in algebra.

Conclusion: The Power of Precise Algebraic Representation

In conclusion, the correct algebraic expression to represent the phrase "Lamar decreased his collection of cards by seven" is x - 7. This expression effectively captures the essence of the phrase by using a variable "x" to denote the initial number of cards and subtracting 7 to represent the decrease. We've explored why this expression is accurate and, equally important, why other options like 7 - x, x + 7, and 7x are incorrect. The analysis of these incorrect options highlights the critical role of precise mathematical representation. Each symbol and operation carries a specific meaning, and misusing them can lead to expressions that do not accurately reflect the intended scenario. The order of operations, the choice of variables, and the selection of appropriate mathematical symbols are all essential components of translating verbal phrases into algebraic expressions. Understanding the concept of variables is fundamental. A variable allows us to represent an unknown quantity, enabling us to create general expressions that can apply to various situations. In this case, "x" represented the initial number of cards, which could be any non-negative integer. The subtraction operation is equally crucial. The phrase "decreased by" directly translates to subtraction, indicating a reduction in quantity. We must ensure that the subtraction is performed in the correct order, subtracting the decrease (7) from the initial quantity (x). This article has emphasized the significance of careful reading and thoughtful analysis when translating verbal phrases into algebraic expressions. It's not just about memorizing rules but about understanding the underlying concepts and applying them accurately. The ability to translate between verbal and algebraic forms is a core skill in mathematics, providing a foundation for problem-solving and mathematical reasoning. By mastering this skill, students can confidently tackle more complex mathematical concepts and real-world applications. The process of breaking down a phrase, identifying its components, and then constructing the corresponding expression is a valuable exercise in critical thinking and mathematical communication. It allows us to express ideas concisely and precisely, making mathematics a powerful tool for understanding and describing the world around us.

Which expression accurately represents the phrase "Lamar decreased his collection of cards by seven"? This question tests your understanding of how to translate verbal expressions into algebraic expressions, a fundamental skill in mathematics. Let's delve into the details and understand the correct representation.

Understanding the Phrase

The phrase "Lamar decreased his collection of cards by seven" indicates a reduction in the number of cards Lamar owns. The key word here is "decreased," which implies a subtraction operation. To represent this mathematically, we need to identify the initial quantity and the amount by which it was decreased.

Identifying the Variable

Since we don't know the initial number of cards Lamar had, we'll represent it with a variable. Let's use "x" to denote the original number of cards in Lamar's collection.

Representing the Decrease

The phrase states that Lamar decreased his collection "by seven." This means we need to subtract 7 from the initial number of cards, which we've represented as "x."

Constructing the Expression

Combining these elements, the algebraic expression that represents the phrase "Lamar decreased his collection of cards by seven" is:

x - 7

This expression accurately reflects the scenario: we start with Lamar's initial number of cards (x) and subtract 7 to account for the decrease.

Evaluating the Options

Now, let's look at the given options and see why only one is correct:

• A. x - 7 (Correct): This is the expression we derived, accurately representing the decrease in Lamar's card collection.
• B. 7 - x (Incorrect): This expression subtracts the initial number of cards (x) from 7. This would be correct if the phrase stated "Seven decreased by Lamar's collection of cards," but that's not what the question asks.
• C. x + 7 (Incorrect): This expression adds 7 to the initial number of cards. The phrase indicates a decrease, not an increase, so addition is not the correct operation.
• D. 7x (Incorrect): This expression represents 7 multiplied by the initial number of cards. Multiplication is not related to the idea of decreasing a quantity by a specific amount.

Conclusion

Therefore, the correct expression to represent the phrase "Lamar decreased his collection of cards by seven" is x - 7 (Option A). This question highlights the importance of carefully interpreting verbal phrases and translating them accurately into algebraic expressions. By understanding the key words and the operations they imply, you can confidently solve similar problems in mathematics.