Visszahelyettesítés

Visszahelyettesítés is a Hungarian term that translates to "back-substitution" in English. It is a fundamental technique used in solving systems of linear equations, particularly after they have been transformed into an upper triangular or row echelon form. The process begins with the last equation in the system, which typically involves only one unknown variable. This variable's value is then calculated directly.

Once the value of the last variable is known, it is substituted into the second-to-last equation. This equation now contains only one unknown variable, whose value can be determined. This iterative process continues, moving upwards through the equations. Each subsequent equation involves one new unknown variable, and the values of the previously determined variables are substituted in to solve for it.

Back-substitution is efficient because it systematically isolates and solves for each variable. It is a crucial step in methods like Gaussian elimination, where the original system of equations is manipulated into an upper triangular form, making back-substitution the straightforward method for finding the solution set. The success of back-substitution relies on the system having a unique solution, which is usually ensured by the form of the matrix after elimination.