    Next: A special case: homogeneous Up: Solution of a system Previous: Solution of a system

## The general case.

The algorithm: Proposition 3.3.1
1.
The system  1 has at least a solution if, and only if, there is no row leading coefficient in the last column.
2.
Suppose that the system  1 has at least a solution. If there exists at least one column without row leading coefficient, then the system has an infinity of solutions.

Example 3.3.2   We solve the system: . Therefore the system has an unique solution given by (x,y,z)=(2,3,3).

Example 3.3.3   We solve the system: . The solution set of the given system is empty.

Example 3.3.4   We solve the system: . We see already that the system has an infinity of solutions. This matrix is associated to the system of equations: The solution set of the given system is: .

Suppose that A is in row-echelon form. An unknown in the column of whom there is a leading coefficient is called a principal unknown; otherwise, it is called a free unknown. In example  3.4, the unknowns x and y are principal, and the unknown z is free. Example 3.3.5   We solve the system: The augmented matrix A of this system is the matrix given in Example  2.6, i.e. The row-reduced matrix B equivalent to A has been computed in Example  2.6: Thus, the given system of equations is equivalent to the following system: The unknowns z and t are principal, as there are pivots in their respective colums. As there is no free unknown on their right, we find constant values for z and t from the two last equations. The y unknown is free, as there is no pivot in its column. By substitution of the values we found for z and t into the first equation, we find an expression for the principal unknown x as a function of y, namely x=2-2y. The set of solutions of the given system of equation is thus:     Next: A special case: homogeneous Up: Solution of a system Previous: Solution of a system
Noah Dana-Picard
2001-02-26