Two methods, one real answer
The system 2x + 3y = 13 and x − y = 1 solves to x = 3.2, y = 2.2 via Cramer's rule. Using substitution instead: the second equation rearranges to x = y + 1, which substituted into the first gives 2(y+1) + 3y = 13, simplifying to 5y − 11 = 0. Feeding that real reduced equation into the linear equation calculator gives y = 2.2 — then x = y + 1 = 3.2, matching the Cramer's-rule answer exactly.
Why turning a 2-variable system into a 1-variable equation always works
Substitution's whole idea is using one equation to express one variable in terms of the other, then plugging that expression into the remaining equation — this eliminates one unknown entirely, turning two equations in two unknowns into exactly the kind of single linear equation the linear equation calculator already solves.
Why the two methods can't actually disagree
Cramer's rule and substitution aren't solving different problems — they're two different procedures applied to the identical pair of equations. Since a system with a unique solution has exactly one (x, y) pair that satisfies both equations simultaneously, any correct method has to land on that same pair.
When substitution is the more practical choice
Cramer's rule is a clean, mechanical formula well-suited to a calculator. Substitution is often more natural by hand when one equation already isolates a variable easily (like x − y = 1 rearranging instantly to x = y + 1) — reaching for whichever method fits the specific system in front of you doesn't change which answer is correct.