Solve the simultaneous equations in real numbers: 6x + 3y = 7xy and 3x + 9y = 11xy. Find the ordered pair (x, y) that satisfies both equations (choose the non-trivial solution if more than one arises).

Introduction:
We are given two equations with product terms (xy), which suggests rearranging to isolate either x or y, or converting to a form that allows elimination. The task is to find the ordered pair (x, y) that satisfies both equations simultaneously. Non-trivial solutions are preferred if multiple solutions arise.

Given Data / Assumptions:

• Equations: 6x + 3y = 7xy and 3x + 9y = 11xy.
• Real-number solutions; trivial (x, y) = (0, 0) may appear but the meaningful solution has nonzero variables.
• Standard algebraic operations are valid.

Concept / Approach:
Treat each equation as linear in one variable after moving the xy-term to one side. Factor where possible, then eliminate using substitution or simultaneous methods. Because xy multiplies both variables, dividing by a variable is permitted only if that variable is nonzero; we keep track of this carefully.

Step-by-Step Solution:

Verification / Alternative check:
Substitute x = 1, y = 3/2 into both equations: (1) 6*1 + 3*(3/2) = 6 + 4.5 = 10.5; 7*1*(3/2) = 10.5. (2) 3*1 + 9*(3/2) = 3 + 13.5 = 16.5; 11*1*(3/2) = 16.5. Both hold exactly.

Why Other Options Are Wrong:
Each alternative fails one (or both) equations upon substitution; signs and magnitudes do not satisfy both identities simultaneously.

Common Pitfalls:
Dividing by zero inadvertently or discarding valid solutions. Always verify candidates in both equations.

Final Answer:
x = 1, y = 3/2