Solve the linear system and compare x and y (use mapping below): I. 4x + 7y = 209 II. 12x − 14y = −38 Mapping: 1 → x > y, 2 → x < y, 3 → x = y, 4 → Relationship cannot be determined.

Introduction / Context:
For two linear equations in two variables, there is a unique solution when the lines are non-parallel. Once x and y are found, the comparison is direct and definitive.

Given Data / Assumptions:

• 4x + 7y = 209
• 12x − 14y = −38

Concept / Approach:
Use elimination by aligning coefficients of y. Add the equations to eliminate y, compute x, then back-substitute to find y. Finally, compare x and y numerically.

Step-by-Step Solution:
From II: 12x − 14y = −38 ⇒ divide by 2 ⇒ 6x − 7y = −19Add with I: (4x + 7y) + (6x − 7y) = 209 + (−19) ⇒ 10x = 190x = 19Back-substitute: 4*19 + 7y = 209 ⇒ 76 + 7y = 209 ⇒ 7y = 133 ⇒ y = 19Hence x = y

Verification / Alternative check:
Plug x = y = 19 into both equations to confirm: 76 + 133 = 209 and 228 − 266 = −38. Both hold.

Why Other Options Are Wrong:
Since x and y are exactly equal, “greater than”, “less than”, or “indeterminate” are incorrect.

Common Pitfalls:
Arithmetic slips when dividing or adding equations. Keep track of signs carefully.

Final Answer:
x = y