Equivalent Expressions For X = 3/4

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In mathematics, understanding equivalent expressions is a fundamental skill that allows us to manipulate equations and simplify problems. This article will delve into the concept of equivalent expressions, specifically focusing on the scenario where x = 3/4. We will examine several expressions to determine which ones are equivalent to 3/16 when this value of x is substituted. By carefully evaluating each expression, we can reinforce our understanding of algebraic manipulation and numerical computation. This comprehensive exploration will not only enhance your mathematical proficiency but also provide a practical approach to solving similar problems in various contexts.

Exploring Equivalent Expressions

In this section, we will dissect each given expression to determine its equivalence to 3/16 when x = 3/4. We will begin by substituting the value of x into each expression and then simplifying the result. This step-by-step process will help us identify which expressions yield the desired outcome. Accurate calculation and simplification are crucial in this exercise. Each expression offers a unique challenge, testing our understanding of different mathematical operations, including addition, subtraction, multiplication, division, and exponentiation. By methodically analyzing each expression, we will develop a clearer understanding of how different mathematical operations interact and influence the final result.

A. 2x + 1/16

To determine if the expression 2x + 1/16 is equivalent to 3/16 when x = 3/4, we need to substitute the value of x into the expression. This means replacing x with 3/4 and performing the necessary calculations. The first step is to multiply 2 by 3/4, which gives us 6/4, or 3/2. Then, we add 1/16 to 3/2. To add these fractions, we need a common denominator, which in this case is 16. Converting 3/2 to a fraction with a denominator of 16, we get 24/16. Now, we can add 24/16 and 1/16, which results in 25/16. Comparing this result to 3/16, we can see that 2x + 1/16 is not equivalent to 3/16 when x = 3/4. This process demonstrates the importance of accurate substitution and simplification in evaluating algebraic expressions.

B. x² - 6/16

Next, let's analyze the expression x² - 6/16. Again, we substitute x = 3/4 into the expression. This means we need to square 3/4 first, which gives us (3/4)(3/4) = 9/16*. Now, we subtract 6/16 from 9/16. The calculation is straightforward: 9/16 - 6/16 = 3/16. Therefore, the expression x² - 6/16 is indeed equivalent to 3/16 when x = 3/4. This example highlights the significance of following the order of operations (PEMDAS/BODMAS) when evaluating expressions. Squaring x before subtracting is crucial to arriving at the correct answer. This meticulous approach ensures accurate evaluation and confirms the equivalence of the expression.

C. (3/8)² ÷ x

For the expression (3/8)² ÷ x, we substitute x = 3/4. First, we square 3/8, which means (3/8)(3/8) = 9/64*. Next, we divide 9/64 by 3/4. Dividing by a fraction is the same as multiplying by its reciprocal, so we multiply 9/64 by 4/3. This gives us (9/64) * (4/3) = 36/192. Simplifying this fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 12. This simplifies 36/192 to 3/16. Thus, the expression (3/8)² ÷ x is equivalent to 3/16 when x = 3/4. This calculation demonstrates the importance of understanding fraction division and simplification. Converting division into multiplication by the reciprocal and simplifying fractions are essential skills in algebraic manipulation.

D. x - 1/4

Finally, let's evaluate the expression x - 1/4. Substituting x = 3/4, we get 3/4 - 1/4. This subtraction is straightforward since the fractions have a common denominator. 3/4 - 1/4 = 2/4. Simplifying 2/4, we get 1/2. Comparing this result to 3/16, we see that x - 1/4 is not equivalent to 3/16 when x = 3/4. This simple subtraction problem reinforces the basic principles of fraction arithmetic. While the calculation itself is uncomplicated, it’s crucial to perform it accurately to determine the expression's equivalence correctly.

Conclusion: Identifying Equivalent Expressions

In summary, by substituting x = 3/4 into the given expressions and simplifying, we found that expressions B. x² - 6/16 and C. (3/8)² ÷ x are equivalent to 3/16. Expression A, 2x + 1/16, and expression D, x - 1/4, are not equivalent to 3/16 for the given value of x. This exercise underscores the importance of careful substitution, accurate calculation, and simplification in determining the equivalence of algebraic expressions. The ability to manipulate expressions and identify equivalent forms is a critical skill in mathematics, applicable in various contexts, from basic algebra to more advanced calculus.

Key Points to Remember:

  • Substitution: Always substitute the value of the variable correctly into the expression.
  • Order of Operations: Follow the order of operations (PEMDAS/BODMAS) to ensure accurate calculations.
  • Fraction Arithmetic: Be proficient in adding, subtracting, multiplying, and dividing fractions.
  • Simplification: Simplify fractions and expressions to their simplest form.
  • Comparison: Compare the simplified result with the target value to determine equivalence.

By mastering these skills, you can confidently tackle a wide range of mathematical problems and gain a deeper understanding of algebraic concepts. This article provides a comprehensive approach to evaluating equivalent expressions, empowering you to succeed in your mathematical endeavors.