If 3x + 5y + 7z = 49 and 9x + 8y + 21z = 126, then what is the numerical value of y in this system of linear equations?

Introduction / Context:
In this aptitude problem we are given two linear equations in the three variables x, y and z, and we are asked to determine only the value of y. The question checks whether you can use elimination in a focused way, without trying to find the entire solution set of the system.

Given Data / Assumptions:
- 3x + 5y + 7z = 49
- 9x + 8y + 21z = 126
- x, y and z are real numbers and we only need the value of y.

Concept / Approach:
The key idea is to eliminate x and z simultaneously. If we can create a situation where the x and z terms cancel when we combine the equations, the remaining equation will involve only y, which can then be solved directly.

Step-by-Step Solution:
Step 1: Multiply the first equation by 3 so that the coefficient of x matches the second equation: 3(3x + 5y + 7z) = 3 * 49.Step 2: This gives 9x + 15y + 21z = 147.Step 3: Now subtract the original second equation (9x + 8y + 21z = 126) from this new equation.Step 4: (9x + 15y + 21z) − (9x + 8y + 21z) = 147 − 126.Step 5: The x and z terms cancel, leaving 7y = 21.Step 6: Divide both sides by 7 to get y = 21 / 7 = 3.

Verification / Alternative check:
We can quickly check consistency. If y = 3, then the first equation becomes 3x + 5(3) + 7z = 49, so 3x + 15 + 7z = 49, or 3x + 7z = 34. The second equation becomes 9x + 8(3) + 21z = 126, so 9x + 24 + 21z = 126, or 9x + 21z = 102. These two equations in x and z are consistent, which confirms that y = 3 is valid.

Why Other Options Are Wrong:
Values like y = 2, 4 or 5 do not satisfy the relationship between the two equations after elimination. Substituting any of these into both equations will lead to inconsistent equations for x and z, so they cannot be correct.

Common Pitfalls:
A common mistake is to think that because there are three variables and only two equations, no single variable can be determined. However, sometimes a specific combination of coefficients makes one variable uniquely determined. Another error is performing elimination incorrectly, such as multiplying or subtracting equations wrongly, which leads to incorrect values of y.

Final Answer:
The correct value of y is 3.