If x - 2y = 2 and 3x + y = 20, then what are the values of the ordered pair (x, y)?

Introduction / Context:
This question involves solving a pair of simultaneous linear equations in two variables. Such systems form the backbone of algebra and appear frequently in aptitude tests to assess a candidate ability to handle multiple equations and find a unique ordered pair solution (x, y).

Given Data / Assumptions:

• First equation: x - 2y = 2
• Second equation: 3x + y = 20
• x and y are real numbers.
• We seek the ordered pair (x, y) that satisfies both equations simultaneously.

Concept / Approach:
We can use either substitution or elimination. The first equation is already simple to express x in terms of y or vice versa. Using substitution is convenient:

• From x - 2y = 2, express x as x = 2 + 2y.
• Substitute this x into 3x + y = 20 and solve for y.
• Then back substitute to get x.

Step-by-Step Solution:
From x - 2y = 2, we get x = 2 + 2y Substitute into 3x + y = 20: 3(2 + 2y) + y = 20 This gives 6 + 6y + y = 20 Combine like terms: 6 + 7y = 20 So 7y = 14, hence y = 14 / 7 = 2 Now find x: x = 2 + 2y = 2 + 4 = 6 Thus the ordered pair is (x, y) = (6, 2)

Verification / Alternative check:
Check (6, 2) in both equations. First: 6 - 2*2 = 6 - 4 = 2, correct. Second: 3*6 + 2 = 18 + 2 = 20, also correct. No other listed option will satisfy both equations when substituted, so (6, 2) is the unique valid answer.

Why Other Options Are Wrong:
For (4, 1), the first equation becomes 4 - 2 = 2 (works) but the second becomes 3*4 + 1 = 13, not 20. For (3, 2), x - 2y = 3 - 4 = -1, not 2. For (5, 5), x - 2y = 5 - 10 = -5, incorrect. For (2, 6), x - 2y = 2 - 12 = -10, also incorrect.

Common Pitfalls:
Candidates sometimes make arithmetic mistakes when substituting or combining like terms, for example miscalculating 6 + 7y or incorrectly dividing 14 by 7. Writing each step clearly and checking substitution into both equations helps prevent such errors.

Final Answer:
The correct ordered pair is (6, 2).