In the xy-coordinate system, the points (a, b) and (a + 3, b + k) both lie on the line defined by x = 3y − 7. Determine the value of k that must hold for the second point to remain on the same line.

Introduction / Context:
In coordinate geometry, if two points lie on the same straight line, each point must satisfy the line’s equation. This question tests your ability to substitute coordinates into a linear equation written in the form x = 3y − 7 and solve for an unknown increment k that keeps a translated point on the same line.

Given Data / Assumptions:

• Line: x = 3y − 7
• Point 1: (a, b) lies on the line
• Point 2: (a + 3, b + k) also lies on the line
• a, b, k are real numbers

Concept / Approach:
If (x, y) lies on x = 3y − 7, then x equals 3y − 7. Apply this for each point. Use the fact that the first point already satisfies a = 3b − 7, then impose the second point to find k by simple algebraic elimination.

Step-by-Step Solution:
From (a, b) on the line: a = 3b − 7.For (a + 3, b + k) on the same line: a + 3 = 3(b + k) − 7.Expand: a + 3 = 3b + 3k − 7.Substitute a = 3b − 7 into the left side: (3b − 7) + 3 = 3b + 3k − 7.Simplify left side: 3b − 4 = 3b + 3k − 7.Subtract 3b both sides: −4 = 3k − 7.Add 7 both sides: 3 = 3k.Therefore, k = 1.

Verification / Alternative check:
Pick any b (say b = 3), then a = 3*3 − 7 = 2. The second point should be (5, 4) when k = 1. Check: x = 5 and 3y − 7 = 3*4 − 7 = 5, which matches. So k = 1 is consistent.

Why Other Options Are Wrong:
k = 3 leads to a + 3 = 3b + 9 − 7, which contradicts a = 3b − 7. k = 9 and k = 7/3 similarly fail the equality. k = 0 would give a + 3 = 3b − 7, which is false since a = 3b − 7 implies a + 3 = 3b − 4, not 3b − 7.

Common Pitfalls:
Mixing the roles of x and y in the equation x = 3y − 7; forgetting to use the first point’s relation to substitute for a; or incorrectly distributing 3 over (b + k).

Final Answer:
1