eliminointimenetelmään

Eliminointimenetelmä, often translated as elimination method or linear combination, is a fundamental algebraic technique used to solve systems of linear equations. The core idea behind this method is to systematically eliminate one or more variables from the system by adding or subtracting multiples of the equations. This process reduces the number of variables, thereby simplifying the system and making it easier to solve for the remaining variables.

To apply the elimination method, one typically aligns the equations vertically. The next step involves choosing a variable to eliminate. This is done by manipulating the coefficients of that variable in the equations so that they are either opposites (e.g., 2x and -2x) or identical. If the coefficients are opposites, the equations are added together. If they are identical, one equation is subtracted from the other. The result is a new equation with one fewer variable. This process is repeated until a single equation with a single variable is obtained, which can then be solved directly. Once the value of one variable is found, it is substituted back into one of the earlier, simplified equations to find the value of another variable, and so on, until all variables in the original system have been determined. The elimination method is particularly efficient for systems with integer coefficients and can be extended to solve systems with any number of variables.