- The algebraic expression \(\frac{3}{4}(8x + 4)\) simplifies to \(6x + 3\).
- To verify this, substitute a common value like \(x = 2\) into both original and simplified expressions.
- Since \(\frac{3}{4}(8 \times 2 + 4) = 15\) and \(6 \times 2 + 3 = 15\), the results match.
- This numerical check confirms the equivalence, identifying option (b) as the correct procedure.
- The final answer is \(\boxed{b}\).
Explanation
- Analyzing the expressions and equivalence
To verify if the simplification of an algebraic expression is correct, we need to check if the original expression and the simplified expression are equivalent for all values of \(x\). The original expression is \(E_1(x) = \frac{3}{4}(8x + 4)\) and the simplified expression is \(E_2(x) = 6x + 3\). We can expand \(E_1(x)\) to see if it matches \(E_2(x)\):
\[E_1(x) = \frac{3}{4} \times 8x + \frac{3}{4} \times 4 = 6x + 3.\]
Since \(E_1(x) = E_2(x)\), the simplification is correct. To verify this result, we need a method that checks for identity.
- Evaluating the verification procedures
Let's evaluate the given options:
- Option (a): Substituting different values (e.g., \(1\) into the first and \(2\) into the second) does not allow for a direct comparison of the two expressions. If the results are different, it tells us nothing about their equivalence because we are comparing them at different input values.
- Option (b): Substituting the same value \(x = c\) into both expressions, \(E_1(c)\) and \(E_2(c)\), and comparing the results is a standard and reliable way to check for errors. If \(E_1(c) = E_2(c)\), it provides strong evidence that the expressions are equivalent. For instance, if we pick \(x = 2\), then \(E_1(2) = \frac{3}{4}(8(2) + 4) = \frac{3}{4}(20) = 15\) and \(E_2(2) = 6(2) + 3 = 15\). Since both yield the same result, it suggests the simplification is correct.
- Option (c): Calculating the difference \(E_1(x) - E_2(x)\) is also a valid algebraic approach. If the expressions are equivalent, the difference must be identically zero for all \(x\). While this is mathematically rigorous, it involves an additional subtraction step that can be error-prone for students compared to a simple numerical substitution.
- Option (d): Adding the expressions would result in a new expression, which does not directly help in verifying their equality.
Given these considerations, substituting the same value into both expressions (Option b) is the most intuitive and widely used procedure for students to quickly verify their algebraic work.
- Conclusion and final answer
Comparing the options, substituting a specific value like \(x=2\) into both expressions (Option b) is a highly practical and standard method for verifying the correctness of an algebraic simplification. It provides an immediate check for consistency, making it the most appropriate procedure for a student to follow to verify their work. Therefore, the correct procedure is option b. The final answer is \(\boxed{b}\).
Examples
In everyday life, verifying your work is just as important as doing the calculation itself. For example, if you are calculating the total cost of items with a discount, you might use two different methods to reach the same conclusion: one by calculating the total first and then applying the discount, and another by applying the discount to each item individually before summing them up. By checking that both methods result in the same final price, you gain confidence that your math is accurate. Similarly, in algebra, testing your simplified expression with a specific number is like checking your bill with two different methods—it ensures you haven't made any mistakes.