If 2x + 6y = 3xy and 10x − 3y = 4xy, what is the ordered pair (x, y) that satisfies both equations?

Introduction / Context:
This question examines your skill in solving a system of two equations with two variables where the variables appear both linearly and as products. Such questions appear in algebra and aptitude exams to test algebraic manipulation and substitution skills.

Given Data / Assumptions:

- Equation 1: 2x + 6y = 3xy
- Equation 2: 10x − 3y = 4xy
- x and y are real numbers
- We are looking for an ordered pair (x, y) that satisfies both equations simultaneously

Concept / Approach:
The equations involve products of x and y, so a convenient method is to rearrange each equation to group terms involving xy on one side and linear terms on the other. Then you can express one variable in terms of the other, or use elimination to solve the system. Since the answer choices are given as simple integer pairs, we expect integer solutions.

Step-by-Step Solution:
Step 1: Rewrite Equation 1 as 2x + 6y = 3xy. Move all terms to one side: 3xy − 2x − 6y = 0.Step 2: Factor Equation 1 by grouping: x(3y − 2) − 6y = 0. This gives x(3y − 2) = 6y.Step 3: Rewrite Equation 2 as 10x − 3y = 4xy. Move all terms to one side: 4xy − 10x + 3y = 0.Step 4: Factor Equation 2 by grouping: 2x(2y − 5) + 3y = 0. While this does not factor nicely into simple linear factors, we can instead check the integer answer choices directly.Step 5: Test option (3, 2) in Equation 1: 2*3 + 6*2 = 6 + 12 = 18, and 3*3*2 = 18, so Equation 1 is satisfied. Test in Equation 2: 10*3 − 3*2 = 30 − 6 = 24, and 4*3*2 = 24, so Equation 2 is also satisfied.Step 6: Therefore, (3, 2) is a valid solution. None of the other integer pairs produce equality in both equations.

Verification / Alternative check:
To be thorough, you can test other options. For example, for (2, 3) in Equation 1: 2*2 + 6*3 = 4 + 18 = 22 while 3*2*3 = 18, so the equation fails. Similar mismatches occur for (4, 6), (6, 4), and (0, 0) in one or both equations. Only (3, 2) satisfies both equations.

Why Other Options Are Wrong:
- (2, 3) fails Equation 1, since 22 is not equal to 18.
- (4, 6) fails because 2*4 + 6*6 = 8 + 36 = 44, but 3*4*6 = 72.
- (6, 4) fails similarly, giving mismatched left and right sides.
- (0, 0) satisfies Equation 1 trivially but fails Equation 2, where 10*0 − 3*0 = 0 while 4*0*0 = 0, but this pair is usually excluded in aptitude contexts, and it is not among realistic non zero choices.

Common Pitfalls:
One common mistake is to try to divide by x or y without considering that they could be zero. Another issue is algebraic errors when factoring or moving terms across the equality sign. A practical exam trick is to quickly test the given answer choices whenever equations look messy, especially when integer solutions are expected.

Final Answer:
The ordered pair (x, y) that satisfies both equations is (3, 2).