Substituutiomenetelmässä

Substituutiomenetelmässä, often translated as the substitution method, is a fundamental technique used in mathematics to solve systems of equations, particularly systems of linear equations. The core principle of this method is to replace one variable in an equation with an expression involving the other variable(s) from a different equation. This process reduces the number of variables in one equation, making it easier to solve.

To apply the substitution method, one typically follows these steps. First, choose one of the equations and solve it for one variable in terms of the other. For instance, if you have equations like x + y = 5 and 2x - y = 1, you could solve the first equation for x, yielding x = 5 - y. Second, substitute this expression into the other equation. Using the example above, you would substitute (5 - y) for x in the second equation: 2(5 - y) - y = 1. This results in an equation with only one variable, which can then be solved. In the example, 10 - 2y - y = 1, leading to 10 - 3y = 1, and thus -3y = -9, so y = 3. Finally, substitute the value found back into either of the original equations, or into the expression derived in the first step, to find the value of the other variable. In our example, substituting y = 3 into x = 5 - y gives x = 5 - 3, so x = 2. Therefore, the solution to the system is x = 2 and y = 3. The substitution method is versatile and can be applied to systems with more than two variables or non-linear equations, though the complexity increases with more variables.