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Chapter 1
Linear Integer Equations

In a small class 12 students sit in chairs that are arranged in rows of equal length. The following rectangular table shows the seating arrangement:
[1] 
		Jim	Sally	Rich	Frances
		Bob	Jack	Rose	Violet
		Joseph	Bill	Ann	June
Each of the three rows has the names of 4 students. But each of the four columns has the names of 3 students. This table as size of 3x4 (3 by 4). It is customary to give the number of rows first, then the number of columns.

In Row 2 are Bob, Jack, Rose, Violet. In Column 4 are Frances, Violet, Bill. So Violet sits in Row 3 and Column 4. This location is denoted as [3,4]. The persons whose name begins with the letter J sit in seats at locations [1,1], [2,2], [3,1], [3,4].

In an intuitive sense, these location numbers in brackets locate the empty seats. If the table in [1] is given the name T then T[2,3] = Rose, T[3,4] = June. T gives the name of the person sitting in chair [ _,_ ] .

The teacher may want to interchange (swap) Row 2 and Row 3 to get a different table:
[2] 

		Jim	Sally	Rich	Frances 
		Joseph	Bill	Ann	June
		Bob	Jack	Rose	Violet
If this table [2] is given the name S, then S[2,3] = T[3,3].

A table may have (the names of) things of a common type placed at all the locations in it. In a table an entry has been placed at a location [1,1], an entry has been placed at [1,2], etc.This creates a table of entries, whatever they are. The most common entries in these discussions are integers and sometimes fractions which are quotients of integers (denominators not zero). For example,
[3] 

		 2   1  -7   4 
	   	 4  -3   1   5
		-2   9   4   2
This is another 3x4 table. Its entries are integers. A table of numbers is called a matrix. If it is desirable to give a name "A" to the matrix, then the notation becomes:
		   2   1  -7   4 
	  A =	   4  -3   1   5
		  -2   9   4   2


Requiring the drawing of smooth curves the common notation for this matrix is given to the right.

The arithmetical nature of a matrix allows their frequent applications to various fields of mathematics and other fields of science. They can be applied in particular to simultaneous linear equations. The word "simultaneous" implies two or more components of an event happening together, and all should be considered together. In simultaneous equations the same unknowns (letters) are used. The word "linear" implies simple unknowns, no exponents, no roots, no logarithms, no trigonometric functions, etc. The two simultaneous linear equations in x and y
[4]

3x + 4y = 10
2x + 5y = 9
can easily be solved: x = 2, y = 1. (For a process of solving these equations click here.) The equations
[5]
3u + 4v = 10,
2u + 5v = 9
have the solution u = 2, v = 1. Other letters can be used for the unknowns. This implies that the letters for the unknowns play a minor role in the process of solving for the values of those unknowns.

Omit the symbols for unknowns and the equal signs in [4] and [5] produces the same 2x3 matrix of the equations:
[6] 

			3   4    10                3x + 4y =  10
			2   5     9                 2x + 5y =   9
Multiply Row 1 by -2 and Row 2 by 3, and replace Row 3 by the result produces the matrix of the equations (in a shorter notation, Row 2 <-- -2(Row 1) + 2(Row 2)):
[7] 
			3   4    10                3x + 4y =  10
			0   7     7                     7y =   7
Now divide the new Row 2 by 2 to get a new matrix:
[8]: 
			3   4    10                 3x + 4y =  10
			0   1     1                       y =   1
Replace Row 1 by -4x(Row 2) + Row 1:
[9] 
		   3   0    6                    3x     =   6
		   0   1    1                       y   =   1
Divide Row 1 by 3 to get:
[10] 
                                    1    0    2                  x      =   2
                                    0    1    1                    y   =   1
The solution is in the right column
[11]
		2                         x = 2
		1                         y = 1

Notice that the row operations on the matrix of the equations correspond to legal operations on the equations.

Three rules of algebra for handling simultaneous liner Equations are useful here:
(E1) In any equation, every number may be multiplied by the same non-zero number.
(E2) Any two equations may be added together and this sum may replace either equation (but not both).
(E3) Any two equations may be interchanged.

The corresponding rules for handling Matrices of equations are:
(M1) In any row, every number may be multiplied by the same non-zero number.
(M2) Any two rows may be added together and this sum may repalce either row (but not both).
(M3) Any two rows may be interchanged.

No similar rules will apply here for the columns.

Applying the above rules changes the equations or matrix of the equations. In these discussions the symbol => will mean "rules for handling rows have been applied to produce." For example [6] - [10]may be written:
[12] 

3   4    10	 =>   3   4    10   =>   3   4    10    =>   3   0    6   =>  1   0    2
2   5     9	      0   7     7        0   1     1         0   1    1       0   1    1
Often the chains showing every conversion are long. Some of the intermediate matrices may be omitted. The equations
[13]
2x + 3y + 4z = 21
5x + 4y + 2z = 44
x + 7y + 6z = 41
start the chain:
[14]
	2   3   4    21              -1   0   0   -4           1   0   0   -4
	5   4   2    44     =>        0  -1   0   -7    =>     0   1   0    7    
	1   7   6    41               0   0   1    2           0   0   1   -2
For more detail click here.

It is quite possible that linear equations cannot be solved in the usual sense. For example, the simple equation
[15]

0x = 0
cannot be solved to give a single answer x = ?. In this case x can be any number. So there are infinitiely many "answers."
The other situation is - no answer at all. For example,
[16]
0x = 3
Zero times any number gives zero. The equation [16] presents an unresolvable contradiction.
In both cases zero plays an inportant role as a coefficient. Incidently, the 1x2 matrices of these two "pathological" equations are (0 0) and (0 3) respectively.

A similar situation may exist with simultaneous equations with 2 or more unknowns. A very obvious example is
[17]

3x - 4y = 5
3x - 4y = 5
Intuitively speaking, the second equation supplies no new information to what has been given by the first equation. So the second equation may be "removed" without any loss of information. Atttempts to solve [17] leads to the statement 0 = 0, suggesting that infinitely many solutions exist. The matrix of the equations has two identical rows.

The 0=0 result can be obtained from the equations
[18]

3x - 4y = 5
6x - 8y = 10
Mulltiplying the first equation by 2 does not provide any more information about the solution. The matrix of these equations shows a bottom row proportional to the top row.

An attempt to solve the following equations leads to the contradiction like 0=1 (twice the first equation subtracted from the second equation):
[19]

3x - 4y = 5
6x - 8y = 11