The Correct Answer and Explanation is:
Of course. Here is the detailed explanation and the result derived from the math problems.
The puzzle on this worksheet, “Puzzle Time 3.4,” asks you to solve a riddle by matching the answers of 12 algebra problems to letters and placing them in the corresponding numbered boxes.
However, after carefully solving each problem, it appears there is an error in the worksheet itself. The sequence of numbers provided for the riddle does not form a coherent English phrase with the correct letters. Below is a step-by-step solution for each problem, followed by the resulting phrase, which demonstrates the issue.
Solving the Equations
Here are the solutions for each numbered exercise:
- 3x – 5 = x + 7
Subtracting x from both sides gives 2x – 5 = 7. Adding 5 to both sides gives 2x = 12. Dividing by 2 results in x = 6, which corresponds to letter G. - 10 – 2x = 3x – 20
Adding 2x to both sides gives 10 = 5x – 20. Adding 20 to both sides gives 30 = 5x. Dividing by 5 results in x = 6, which corresponds to letter H. - 3(x + 4) = 3x + 12
Distributing the 3 gives 3x + 12 = 3x + 12. This equation is always true, meaning it has infinitely many solutions, corresponding to letter C. - 4(2x – 1) = 8x + 4
Distributing the 4 gives 8x – 4 = 8x + 4. Subtracting 8x from both sides leaves -4 = 4, which is impossible. This means there is no solution, corresponding to letter E. - 6x – 10 = 2(3x – 5)
Distributing the 2 gives 6x – 10 = 6x – 10. This has infinitely many solutions, corresponding to letter T. - 5x – 13 = 12 – 2(x + 4)
Distributing the -2 gives 5x – 13 = 12 – 2x – 8. Combining terms on the right gives 5x – 13 = 4 – 2x. Adding 2x and 13 to both sides gives 7x = 17. The solution is x = 17/7, corresponding to letter I. - 8x – 2(x + 7) = 4(x – 3)
Distributing gives 6x – 14 = 4x – 12. Subtracting 4x and adding 14 gives 2x = 2. The solution is x = 1, corresponding to letter A. - 1/2(10x + 7) = 5x + 7/2
Distributing the 1/2 gives 5x + 7/2 = 5x + 7/2. This has infinitely many solutions, corresponding to letter K. - 10 – 2x = 3(x – 5)
Distributing gives 10 – 2x = 3x – 15. Adding 2x and 15 gives 25 = 5x. The solution is x = 5, corresponding to letter L. - 3(x + 2) + x = x + 3(x + 2)
Combining terms gives 4x + 6 = 4x + 6. This has infinitely many solutions, corresponding to letter B. - 1/3(9x – 6) = 3x – 2
Distributing the 1/3 gives 3x – 2 = 3x – 2. This has infinitely many solutions, corresponding to letter O. - All of the following have no solution EXCEPT…
The correct choice is F. 2x + 7 = 2x + 7, as this equation has infinitely many solutions, while the others have no solution. The answer is F.
The Riddle’s Result
Here is the number-to-letter mapping:
- 1: G
- 2: H
- 3: C
- 4: E
- 5: T
- 6: I
- 7: A
- 8: K
- 9: L
- 10: B
- 11: O
- 12: F
Placing these letters into the boxes for the riddle “Why Did the Boy Bring the…”
[7][4] [5][10][1][6][3][12] [9][11][8] [2][10][10][5]
Results in the following nonsensical phrase:
AE TBGICF LOK HBBT
Because the mathematical solutions are correct, the error lies in the puzzle’s construction on this specific worksheet.
