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Examples of linear systems

1 - Solve the system img.

The augmented matrix A|b of the system is: img,

which is row equivalent to the reduced row-echelon matrix B|c: img.

The solution of the system is easily found: img. The other unknowns are completely arbitrary. This system has n=6, m=4, r=3, and so ∞6-3=∞3 solutions.

2 - Solve the system img, where t is a real number.

The augmented matrix and its reduced form are: img. The system is consistent if and only if t=2. In this case the reduced row-echelon form of the matrix is: img. We read off the unique solution of the system: img. From a geometrical point of view this result can be interpreted as follows: given two non parallel straight lines and a third variable straight line, find the value of t for which this third line has a unique point in common with the preceding two.

3 - Solve the system img, where s and t are real numbers.

Write the augmented matrix A|b: img. Now start applying Gauss-Jordan algorithm. img  img; img.

Now we must distinguish between to cases:

first published on march 15 2002 - last updated on september 01 2003