If the variables x and y satisfy the simultaneous equations 7x + 6y = 5xy and 10x - 4y = 4xy, then what are the values of x and y?

Introduction / Context:
Simultaneous equations involving products of variables occur often in aptitude tests, especially when modelling word problems or algebraic conditions. In this question, both equations contain terms of the form xy as well as linear terms in x and y. Your task is to solve the system and determine the pair (x, y) that satisfies both equations. This tests your ability to manipulate and simplify non linear simultaneous equations.

Given Data / Assumptions:

• The equations are 7x + 6y = 5xy and 10x - 4y = 4xy.
• x and y are real numbers, and xy appears in both equations.
• We are interested in solutions that match the given options.
• We ignore trivial solutions that make denominators zero if we divide by x or y.

Concept / Approach:
A useful approach is to rearrange each equation so that terms involving xy are on one side and linear terms on the other, and then factor or divide to find relationships between x and y. Another way is to treat the equations as linear in x when y is considered a parameter, or linear in y when x is considered a parameter, and then solve them systematically. We can also look for integer solutions among the options by substitution, which is efficient when the answer choices are discrete pairs of small integers.

Step-by-Step Solution:
Step 1: Start with the first equation: 7x + 6y = 5xy. Rearrange to 5xy - 7x - 6y = 0. Step 2: Factor by grouping: 5xy - 7x - 6y = x(5y - 7) - 6y. To factor further, note that we can write it as (5y - 7)x - 6y. Step 3: Similarly, rewrite the second equation 10x - 4y = 4xy as 4xy - 10x + 4y = 0, or 4xy - 10x + 4y = x(4y - 10) + 4y. Step 4: Instead of complex factoring, test the likely integer solution from the options. Try x = 4, y = 2. Step 5: Substitute into the first equation: 7x + 6y = 7*4 + 6*2 = 28 + 12 = 40, and 5xy = 5*4*2 = 40, so the first equation is satisfied. Step 6: Substitute into the second equation: 10x - 4y = 10*4 - 4*2 = 40 - 8 = 32, and 4xy = 4*4*2 = 32, so the second equation is also satisfied.

Verification / Alternative check:
You can verify that other listed options do not work. For example, with x = 3 and y = 2, the left side of the first equation is 7*3 + 6*2 = 21 + 12 = 33, while the right side is 5*3*2 = 30, so the equation is not satisfied. Similar checks show that the other pairs fail at least one equation. Thus, x = 4 and y = 2 is the only valid pair among the choices, confirming our solution.

Why Other Options Are Wrong:

• Option a (x = 3, y = 2) does not satisfy the first equation, as 33 is not equal to 30.
• Option b (x = 2, y = 3) fails because 7*2 + 6*3 = 14 + 18 = 32 is not equal to 5*2*3 = 30.
• Option d (x = 5, y = 6) produces mismatched values in both equations due to large products.
• Option e (x = 0, y = 0) makes the left sides zero but also collapses the equations; however, such a trivial pair usually does not match typical aptitude expectations and is not consistent with the derived relations when dividing by x or y.

Common Pitfalls:
Many students try to solve such systems entirely symbolically and get lost in algebraic manipulation. Others may forget that testing integer options is a valid and efficient strategy when multiple choice answers are given. A common error is also forgetting to check the solution in both equations. Working systematically, performing substitutions carefully, and verifying each candidate pair prevent these issues and lead to a reliable answer.

Final Answer:
The values of the variables are x = 4, y = 2.