# 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 x*, x*_{1}*, ..., x*_{2}*as an equation that can be expressed in the form of:*_{n}
Where a

*, a*_{1}*, ..., a*_{2}*and b are real numbers.*_{n}**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 x

*, x*_{1}*, ..., x*_{2}*are unknown variables while a*_{n}*and b*_{ij}*all say real numbers.*_{i}
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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