Solve the simultaneous equations 4x - 3y = 10 and 3x + y = 14 and identify the correct ordered pair (x, y).

Introduction / Context:
This problem involves solving a system of two linear equations in two variables. Such questions are fundamental in algebra and aptitude tests and model real life scenarios where two linear conditions must be satisfied simultaneously by the same pair of values.

Given Data / Assumptions:

• Equation 1: 4x - 3y = 10
• Equation 2: 3x + y = 14
• We must find the ordered pair (x, y) that satisfies both equations.

Concept / Approach:
To solve simultaneous linear equations, standard methods include substitution, elimination, or matrix methods. For two simple equations, the elimination method is very efficient. We eliminate one variable by combining the equations so that its coefficient becomes zero, then solve for the remaining variable and back substitute.

Step-by-Step Solution:
From Equation 2, express y in terms of x: 3x + y = 14 implies y = 14 - 3x. Substitute this into Equation 1: 4x - 3(14 - 3x) = 10. Simplify: 4x - 42 + 9x = 10, so 13x - 42 = 10. Add 42 to both sides: 13x = 52. Divide by 13: x = 52 / 13 = 4. Now substitute x = 4 back into y = 14 - 3x. y = 14 - 3*4 = 14 - 12 = 2. Therefore the solution is (x, y) = (4, 2).

Verification / Alternative check:
Check in both equations: Equation 1: 4*4 - 3*2 = 16 - 6 = 10, which is satisfied. Equation 2: 3*4 + 2 = 12 + 2 = 14, which is also satisfied. Since the pair satisfies both equations, it is the unique solution for this system.

Why Other Options Are Wrong:
(5, 3) gives 4*5 - 3*3 = 20 - 9 = 11, not 10, so it fails the first equation. (11, 1) fails the second equation because 3*11 + 1 = 34, not 14. (2, 3) fails both equations as 4*2 - 3*3 = 8 - 9 = -1 and 3*2 + 3 = 9. (6, 4) gives 24 - 12 = 12 for the first equation, not 10, so it is incorrect.

Common Pitfalls:
Typical mistakes include arithmetic errors when distributing or combining like terms, or substituting incorrectly. Another common issue is mixing up x and y when checking ordered pairs. Writing the substitution step clearly and checking the final candidate in both equations avoids such errors.

Final Answer:
The correct ordered pair is (4, 2).