Introduction of Linear Equation Systems

An algebraic line in the xy plane can be expressed by a form equation,


This kind of equation is called a linear equation in the variables x and y. More generally, we define a linear equation in n variables x1, x2, ..., xn as an equation that can be expressed in the form of:

Where a1, a2, ..., an and b are real numbers.

Example 1,
The following are linear equations:

Note that a linear equation does not involve something that results or the root variable. All variables are only up to the first number and do not appear as arguments for trigonometric functions, logarithmic functions, or for exponential functions.

The following is not a linear equation:



An arbitrary system consisting of m linear equations and n unknown numbers written as


Where x1, x2, ..., xn are unknown variables while aij and bi all say real numbers.
The set of solutions from a linear system is called its solution set.

Example 2,
Look for a set of solving solutions from the linear equation below,


To find a solution to the equation, it is clear from the last two equations, x2 = 3 and x3 = 2. By using these two values ​​in the first equation, x1 = -2 will be obtained.


So, the set of solutions for the system is {(-2, 3, 2)}.

Example 3,

If the 2nd equation is divided by 2, then the system of linear equations will be


These two equations clearly contradict each other. Thus, the system of linear equations in Example 3 does not have a solution. The linear equation system that does not have a solution is said to be inconsistent, whereas a linear equation system that has at least one solution is called consistent.

A system of inconsistent linear equations, the set of solutions is an empty set.
Consider the following system of linear equations,


Both of these equations can be seen as two straight lines in the xy plane, call. Geometrically, the second solution of the equation has 3 possibilities, as shown in the following figure:


In this case:
(i) The linear equation system has no solution
(ii) The linear equation system has exactly one solution
(iii) The linear equation system has many solutions

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