Trivial solution意思

"Trivial solution" is a term used in mathematics, particularly in the context of solving equations or systems of equations. A trivial solution refers to a solution that is the simplest or most obvious one, often because it involves one or more variables being set to zero or to a constant value, such as 0, 1, or -1.

For example, consider the equation:

x^2 - 4x + 3 = 0

One trivial solution to this equation is x = 3, because when you substitute x = 3 into the equation, the left side becomes (3)^2 - 43 + 3 = 9 - 12 + 3 = 0, making the equation true. Similarly, x = 1 is also a trivial solution because when you substitute x = 1, the equation becomes (1)^2 - 41 + 3 = 1 - 4 + 3 = 0.

In the context of systems of equations, a trivial solution might involve all variables being set to zero. For instance, consider the system of equations:

x + 2y = 3 x - y = 1

A trivial solution to this system is x = 0 and y = 0, because when you substitute these values into the equations, both equations become true.

Trivial solutions are often of less interest in mathematical analysis because they do not provide insight into the behavior of the system or the properties of the variables when they are not at their simplest values. However, they are still considered valid solutions and are sometimes used as starting points for more complex analyses or as part of the study of the stability of a system.