MATH.8:5.3 - Permuting coefficients changes which equation was solved

Let the data be coefficients a=(2,1) and right-hand side 5. The equation is 2*x1+x2=5, with known solution x=(1,3).

The swap sends a to (1,2) and x to (3,1). Their scalar product remains 5 because both positions are exchanged. Thus the transformed solution satisfies x1+2*x2=5.

Keeping the original coefficients instead gives 2*3+1=7. The swap preserves the relation between transformed data and transformed solutions; it does not preserve this fixed original equation.

For comparison, x1+x2=5 has equal coefficients. Its data are fixed by the swap, so a solution (1,4) gives another solution (4,1) of the same equation. Neither the swap nor the equation singles out one of them. A requirement for one distinguished answer needs an additional criterion or a compatible choice method.

For a representative-data calculation, take coefficients (1,2) and its solution (3,1). The swap maps those data back to (2,1) and the solution back to (1,3), recovering the original answer. With the equal coefficients (1,1), both identity and swap return the same data, while the chosen solution (1,4) returns as either (1,4) or (4,1). Each is a valid solution, but this choice does not define a transformation-independent rule. This ambiguity concerns the chosen pair; a request for a swap-fixed solution can instead use (5/2,5/2).