Some equations are solved on the basis of a certain system. Find the correct…

Concept

In these letter/number puzzles the result on the right is produced by a fixed rule applied to the two numbers on the left. A very common rule is the difference of two squares: for a pair of numbers x and y, the output equals x2 − y2. Always test a candidate rule against EVERY given equation before trusting it.

Application

Apply x2 − y2 to the two solved rows, then to the unknown:

1. Row with 8 and 6: 82 − 62 = 64 − 36 = 28. This matches the given 28.

2. Row with 11 and 5: 112 − 52 = 121 − 25 = 96. This matches the given 96.

3. Unknown row with 15 and 7: 152 − 72 = 225 − 49 = 176.

Cross-check

The rule reproduces both supplied results exactly, so it is the governing system. Using the factored form (x − y)(x + y) gives the same answer: (15 − 7)(15 + 7) = 8 × 22 = 176. Hence ? = 176.