Sum–difference to product (two-number system): The sum of two numbers is 25 and their difference is 13. Keeping both conditions true, determine the product of the two numbers.

Introduction / Context:
This question tests solving a two-variable linear system built from a given sum and difference, and then using the recovered numbers to compute their product. It reinforces the standard method for reconstructing two unknowns when their sum and difference are known.

Given Data / Assumptions:

• x + y = 25
• x − y = 13
• x and y are real numbers; no additional constraints are needed.

Concept / Approach:
With the pair of linear equations, the variables can be found by adding and subtracting the equations. Once x and y are known, compute x * y. This is a classic system solution and product evaluation task.

Step-by-Step Solution:
Add the equations: (x + y) + (x − y) = 25 + 13 → 2x = 38 → x = 19Substitute in x + y = 25: 19 + y = 25 → y = 6Product = x * y = 19 * 6 = 114

Verification / Alternative check:
Check the difference: 19 − 6 = 13 (matches). The product computed from the validated pair is therefore correct.

Why Other Options Are Wrong:
104, 315, 325, and 156 are products of other pairs that do not simultaneously satisfy sum 25 and difference 13.

Common Pitfalls:
Mixing up which variable is larger or trying to guess pairs instead of solving systematically. Always use add/subtract to avoid mistakes.

Final Answer:
114