This research explores a family of linear systems where the coefficients and constants follow an arithmetic progression:
(n)x + (n + 1)y = n + 2
(n + 3)x + (n + 4)y = n + 5
At first glance, the solution seems like it should depend on n, since every term changes with it. However, when the system is solved, all the n-terms cancel out during simplification. The result is surprising:
The result is surprising:
(x, y) = (-1, 2)
This solution remains the same for all real values of n, including positive, negative, and decimal values.
The research also examines a ratio formed from the numerators and denominators of the solutions. That ratio consistently evaluates to:
R = 0.5
Just like the solution itself, this ratio does not change with n.
Even though the equations shift with the parameter, the outcome stays constant. This highlights a simple but interesting structural pattern within this family of linear systems.