Equivalent Expressions Explained: Find The Correct Pair

In the realm of mathematics, particularly algebra, understanding equivalent expressions is crucial for simplifying equations, solving problems, and grasping more advanced concepts. Equivalent expressions are mathematical statements that, despite their different appearances, hold the same value for all possible values of the variables involved. This article delves into the process of identifying equivalent expressions, focusing on the given options and explaining the underlying mathematical principles.

Understanding Equivalent Expressions

Before diving into the specific problem, it's essential to define what equivalent expressions truly mean. Two expressions are considered equivalent if they produce the same result when any value is substituted for the variable. This equivalence is achieved through the application of mathematical properties, such as the distributive property, the commutative property, and the associative property. The distributive property, in particular, plays a significant role in simplifying expressions involving parentheses, which is the core of the given problem.

The distributive property states that for any numbers a, b, and c, the following holds true: a(b + c) = ab + ac. This means that the number outside the parentheses (a) is multiplied by each term inside the parentheses (b and c). Applying this property correctly is key to determining if two expressions are equivalent. It is also important to remember the order of operations (PEMDAS/BODMAS), which dictates that we perform operations in the following order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Analyzing the Given Options

The question presents four options, each consisting of a pair of expressions. Our task is to determine which pair represents equivalent expressions. Let's examine each option methodically, applying the distributive property and simplifying the expressions.

Option A: 2(2/5 x + 2) = 2 2/5 x + 1

To analyze this option, we need to apply the distributive property to the left-hand side of the equation. Multiplying 2 by each term inside the parentheses, we get:

2 * (2/5 x) + 2 * 2 = (4/5)x + 4

Now, let's compare this simplified expression with the right-hand side of the equation, which is 2 2/5 x + 1. Converting the mixed number 2 2/5 to an improper fraction, we get 12/5. So the right-hand side becomes (12/5)x + 1. Clearly, (4/5)x + 4 is not equal to (12/5)x + 1. Therefore, option A is incorrect.

The key mistake in option A lies in the incorrect application or misunderstanding of the distributive property and potentially in the handling of mixed numbers. The left-hand side, when correctly distributed, yields (4/5)x + 4, while the right-hand side remains significantly different, indicating a fundamental inequivalence between the two expressions.

Option B: 2(2/5 x + 2) = 4/5 x + 4

Again, we start by applying the distributive property to the left-hand side of the equation:

2 * (2/5 x) + 2 * 2 = (4/5)x + 4

Comparing this with the right-hand side, which is 4/5 x + 4, we see that the two expressions are identical. This means that the expressions are equivalent. Therefore, option B is the correct answer.

Option B accurately demonstrates the principle of equivalent expressions. The correct application of the distributive property on the left-hand side directly results in the expression on the right-hand side, confirming their equivalence. This highlights the importance of precise arithmetic and the proper use of algebraic properties in simplifying and comparing expressions.

Option C: 2(2/5 x + 4) = 4/5 x + 2

Applying the distributive property to the left-hand side of the equation, we get:

2 * (2/5 x) + 2 * 4 = (4/5)x + 8

Comparing this with the right-hand side, which is 4/5 x + 2, we see that the constant terms are different. The left-hand side has a constant term of 8, while the right-hand side has a constant term of 2. This indicates that the expressions are not equivalent. Therefore, option C is incorrect.

The discrepancy in Option C arises from the incorrect simplification after distribution. The left-hand side correctly expands to (4/5)x + 8, but this is not equal to the right-hand side, (4/5)x + 2. This illustrates the necessity of careful calculation and attention to detail when applying mathematical operations to ensure the equivalence of expressions.

Conclusion

Through a methodical analysis of each option, applying the distributive property and simplifying the expressions, we have determined that option B, 2(2/5 x + 2) = 4/5 x + 4, is the correct answer. This pair of expressions demonstrates equivalence, as both sides of the equation yield the same result when simplified. Understanding equivalent expressions is a fundamental skill in algebra, and this problem highlights the importance of accurately applying mathematical properties to identify them.

Which of the following pairs represents equivalent mathematical expressions?

Equivalent Expressions Explained: Find the Correct Pair