To solve small systems where the number of unknowns and equations is less than or equal to 3 without the need for a computer.
There are three different ways:
1. Graphic method.
2. Cramer's rule.
3. Elimination of unknowns
Graphic method.
This method is to plot the equations become the equations leaving a variable in function of another variable. Is important clear the same variable in both equations.
By plotting the equations on the same plane, the intersection of those lines are the values of the variables.
Example:
x-3y=1
x+4y=8
x+4y=8
From:Métodos Numéricos Grupo O1, subgrupo 1, UIS
There is one drawback with respect to the graphical method and it happens when the system is composed of more than three equations, since for three-dimensional graphics would make the cut and where the three lines would be the solution but for more than three the method does not work, this reason loses importance and is limited to solve larger systems.
Although the method gives information on how they behave systems, for example in the following graphs analyze what happens.
In this case happen to be parallel lines and therefore have no solution.
In this case the lines intersect at all points at which the system will have infinitely many solutions.
This special case is called ill-conditioned systems and to the point where the lines intersect is not very clear and can cause problems when developing the system.
Cramer's rule
This rule tells us that each value of the unknowns can be expressed as the division of the determinant of the matrix modified origin, where the vector of responses b is replaced by the position of the unknown to find the array origin, between the determinant of the initial matrix, as follows:
Example:
Second Part.
Elimination of unknowns
In the method for elimination, as its name says, we must remove one of the unknowns, are these, for example: x, y, z, this is achieved by multiplying one of the equations by a number either positive or negative where all components of the equation are affected by this number, and since the sum by the other equation with the unknown excess selected will be deleted.
After eliminated the unknown, the system reduces to a single unknown (if it is a 2X2), which clears and you get the value of the variable. Since this value is to replace the variable in one of the original equations and solve for the missing variable.
Example:
This method can be developed for larger systems but the difficulty and the tedious calculations do unpopular.
Bibliografy:
- Esteven Chapra, Metodos numericos para ingenieros, Quinta Edicion.
- Analisis numerico escrito por richard l. Burden.+
- http://www.mitecnologico.com/Main/SolucionMetodoDeEliminacion
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