eliminointimenetelmällä

Eliminointimenetelmä, often translated as the elimination method or method of elimination, is a technique used in mathematics, particularly in linear algebra and solving systems of equations. Its primary goal is to simplify a system of equations by systematically removing variables. This process typically involves combining equations in such a way that one or more variables cancel out, leading to a simpler system that is easier to solve.

The core idea behind the elimination method is based on the property that adding or subtracting equal quantities from both sides of an equation maintains the equality. When applied to a system of linear equations, this means we can add or subtract multiples of one equation from another without changing the solution set of the entire system. For instance, if we have two equations, say 2x + y = 5 and x - y = 1, we can add the two equations together. The 'y' terms will cancel out (-y + y = 0), resulting in a single equation with only 'x': 3x = 6. This simplified equation can then be solved for 'x'.

Once a variable is eliminated and solved for, its value can be substituted back into one of the original or intermediate equations to find the value of another variable. This back-substitution process continues until all variables in the system have been determined. The elimination method is particularly useful for systems with three or more variables, where graphical methods become impractical and substitution might become cumbersome. It is a fundamental algorithm for solving linear systems, forming the basis for more advanced techniques like Gaussian elimination.