Systems of Linear Equations

Example1:

A certain three digit number N, equals fifteen times the sum of its digits. If its digits are reversed, the resulting number exceeds N by 396. The one's digit is one larger than the sum of the other two. Find N.

Try to find the system of equations that needs to be solved and then look at the Solution using Maple.

Example2:

Find the equation of the parabola that passes through the points (-1,6) (1,4) and (2,9). Solution

The previous two examples show instances of systems of linear equations with a unique solution. This is not always the case. A system of linear equations could have no solution (e.g. when two or more equations are incompatible) or an infinite number of solutions (e.g. when there are more unknowns than equations or some of the equations are redundant).

The Augmented Matrix of a System of Linear Equations

A matrix is a rectangular array of numbers. Matrices have many applications. In particular, a matrix can be used to encode a system of linear equations. For example consider the system of linear equations associated to Example1 above:

	100 x + 10 y +     z  - N = 0

	 15 x + 15 y +  15 z  - N = 0

	    x + 10 y + 100 z  - N = 396

	   -x -    y +     z      = 1

It can be encoded by the block of numbers "A".


A := matrix(4,5,[100,10,1,-1,0, 15,15,15,-1,0, 1,10,100,-1,396, -1,-1,1,0,1]);

                              [ 100  10   1   -1   0  ]
                              [                       ]
                              [  15  15   15  -1   0  ]
                         A := [                       ]
                              [  1   10  100  -1  396 ]
                              [                       ]
                              [  -1  -1   1    0   1  ]

This is the so called, augmented matrix associated to the system above. This is a matrix of 4 rows and 5 columns (or 4 by 5 matrix). It contains all the useful information about the system of linear equations. The rest i.e. the x,y,z,N's, equal signs, and + signs can always be recovered from the matrix if needed. Here are some examples.