Difficulty: Easy
Correct Answer: 23/2
Explanation:
Introduction / Context:
This problem is a straightforward application of linear equations and averages. The expression 2^3 x + 2^3 y means 2 raised to the power 3 multiplied by x plus the same factor multiplied by y, which simplifies to 8x + 8y. From this relation, you are asked to compute the average of x and y, that is (x + y)/2.
Given Data / Assumptions:
Concept / Approach:
First simplify the equation by factoring out the common coefficient 8. This will give an equation in terms of x + y. Once x + y is known, divide by 2 to obtain the average. The steps involve basic algebraic manipulation and no complicated operations.
Step-by-Step Solution:
Start from 8x + 8y = 184.
Factor 8 out of the left hand side: 8(x + y) = 184.
Divide both sides by 8 to isolate x + y: x + y = 184 / 8.
Compute 184 / 8 = 23, so x + y = 23.
The average of x and y is (x + y)/2 = 23/2.
Verification / Alternative check:
If x + y = 23, you can express y as 23 - x and substitute back into 8x + 8y = 184 to confirm that the equation holds for all such pairs (x, 23 - x). This shows that the average is determined uniquely even though x and y individually are not.
Why Other Options Are Wrong:
Option b ( 23/4 ) corresponds to dividing 23 by 4 instead of 2. Option c ( 23 ) is the sum x + y, not the average. Option d ( 11 ) would match an incorrect computation such as 184 / 16. Option e ( 5/2 ) is unrelated to the given numbers and could result from arbitrary guessing.
Common Pitfalls:
One common mistake is to misinterpret 2^3 x as 2^(3x) and thereby complicate the equation unnecessarily. Another pitfall is to forget to divide by 2 when computing the average and to stop at x + y instead. Carefully simplifying the powers and correctly applying the definition of average avoids these errors.
Final Answer:
The average of x and y, given 8x + 8y = 184, is 23/2.
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