Two numbers satisfy the following system: 2 times the first number plus 3 times the second number equals 36, and 3 times the first number plus 2 times the second number equals 39. What is the difference between the two numbers?

Introduction:
This question tests solving a pair of linear equations in two variables and then computing a derived value (the difference). A common strategy is elimination: multiply equations to align one variable, subtract to eliminate it, then back-substitute. Since the coefficients are small, the arithmetic remains clean. After finding both numbers, take their absolute difference to match the question’s wording.

Given Data / Assumptions:

• Let first number = a
• Let second number = b
• 2a + 3b = 36
• 3a + 2b = 39
• We need |a - b|

Concept / Approach:
Use elimination to solve for b first (or a). Multiply the first equation by 3 and the second by 2 so that the a-coefficient becomes equal (6a). Subtract to eliminate a, solve for b, then substitute back to find a. Finally compute the difference between the two numbers.

Step-by-Step Solution:
2a + 3b = 36 ...(1) 3a + 2b = 39 ...(2) Multiply (1) by 3: 6a + 9b = 108 Multiply (2) by 2: 6a + 4b = 78 Subtract: (6a + 9b) - (6a + 4b) = 108 - 78 5b = 30 => b = 6 Substitute into (2): 3a + 2*6 = 39 => 3a + 12 = 39 3a = 27 => a = 9 Difference = |a - b| = |9 - 6| = 3

Verification / Alternative check:
Check in (1): 2*9 + 3*6 = 18 + 18 = 36, correct. Check in (2): 3*9 + 2*6 = 27 + 12 = 39, correct. So the difference is reliable.

Why Other Options Are Wrong:
They are incorrect differences that would occur if one equation is solved incorrectly, elimination is done with wrong multipliers, or subtraction errors occur.

Common Pitfalls:
Mixing up which equation to multiply, subtracting in the wrong order, or forgetting to take the difference after finding the values of a and b.

Final Answer:
The difference between the two numbers is 3.