Comparison of x and y from two equations (use mapping below): I. 8x^2 + 6x = 5 II. 12y^2 − 22y + 8 = 0 Mapping: 1 → x > y, 2 → x < y, 3 → x = y, 4 → Relationship cannot be determined.

Introduction / Context:
When multiple roots exist for both variables, the relation between x and y may vary across admissible pairs. We conclude “indeterminate” if both equality and strict inequality occur among the valid combinations.

Given Data / Assumptions:

• I: 8x^2 + 6x − 5 = 0
• II: 12y^2 − 22y + 8 = 0

Concept / Approach:
Find roots of both equations, then compare. A single consistent relation is needed to declare x > y, x < y, or x = y; otherwise choose “Relationship cannot be determined.”

Step-by-Step Solution:
I: D = 6^2 + 4*8*5 = 36 + 160 = 196 = 14^2x = [−6 ± 14]/16 ⇒ x ∈ {0.5, −1.25}II: Divide by 2 ⇒ 6y^2 − 11y + 4 = 0, D = 121 − 96 = 25 = 5^2y = [11 ± 5]/12 ⇒ y ∈ {4/3 ≈ 1.333, 1/2 = 0.5}Comparisons: x = 0.5 with y = 0.5 gives equality; x = −1.25 with y = 0.5 gives x < y.

Verification / Alternative check:
Since both equality and strict inequality (x < y) occur, no single relation always holds across all valid pairs.

Why Other Options Are Wrong:
Choosing x > y ignores actual values; choosing x = y ignores the other x-root; choosing x < y ignores the equality case.

Common Pitfalls:
Overlooking repeated roots or failing to consider every possible pair when deciding the relation.

Final Answer:
Relationship cannot be determined