A system of linear equations consists of two or more equations with same variables.A system of equations is also called simultaneous equations as any solution to the system should satisfy all the equations in the system at the same time.The equations cannot be solved independently of one another.
Many real world problems are solved modeling the problem in to a system of linear equations in two variables. Any ordered pair which makes all the equations true is called a solution of the system.

Classification of system of linear equations

A system of linear equations is classified into two types based on the solvability of the equation.
  1. Consistent System: A consistent system has either a unique solution or infinitely many solutions.
  2. Inconsistent System : An inconsistent system does not a solution.
Consistent systems with unique solutions are called independent systems while systems with many solutions are called dependent systems.

Methods of solving a system of linear equations

The common methods which are applied solving a system of linear equations are
  1. Substitution method
  2. Elimination method
  3. Graphical method
  4. Applying Cramer's rule
  5. Matrix row reduction methods.
Out of the above mentioned methods, the first three methods are generally used for solving a system of equations in two variables. Computer programs used for solving systems of more number of variables make use of matrix row reduction methods.

Solving Systems of linear equations by substitution

A system of linear equations in two variables generally consists of two equations.  Using one of the equations, one of the variables is solved literal in terms of the other variable. Then this expression is substituted for the solved variable in the second equation rendering it to an equation in one variable.  The resulting equation is then solved applying simplifications steps and using undoing rules for equality. This method is very efficient when one of the equations is given solved for one of the variables or the equation contains a variable whose coefficient is 1.
Examples:
Consider the two system of equations,
1. $y=2x-3$                                                           2  .$a+2b = 7$
   $3x-y=4$                                                                $3x-2b= 5$
In the first system, the first equation is given solved for y. Hence the expression can be substituted in the second equation and the resulting equation can be solved for x. The solution of x so obtained can be back substituted in equation (1) to get the value of y.

In the second equation, the first equation can be solved in one step for a (note the coefficient of a is 1). The resulting expression can be substituted in the second equation to get an equation purely in b.

Solving systems of linear equation using elimination

This method is also called the addition method for solving system of equations as this method uses addition or subtraction property of equality for eliminating one of the variables. The coefficients of one of the variable are made equal in both the equations by suitably multiplying the two equations.  The two equations are then added or subtracted to get rid of this variable.

Examples:
Consider the two systems of equations
1.  $2x+5y=17$                                                      2.  $3a+2b=1$
     $6x-5y=-9$                                                         $4a+3b=-2$
In the first system the coefficients of the variable y is equal and opposite in the two equations.  Hence y can be eliminated by adding the two equations. The resulting equation can be solved for x. The value of y can be got by back substituting the value of x solved in any one of the equations.

In the second equation either a or b can be eliminated. For eliminating the variable a, the coefficient of a is made as 12(LCM of 3 and 4) in both the equations by suitably multiplying the equations. Then the variable a can eliminated by subtracting one equation from the other.

Solving systems of linear equations by graphing

The graph of a linear equation is a straight line in coordinate plane.  Hence the solution of the system is the point of intersection of the graphs of the equations, which is a point common to both the lines.
If the line drawn intersect at a point, then the system has a unique solution.
If the lines drawn are parallel we can conclude that the system is inconsistent and does not have a solution.
If the lines coincide, then the system is consistent with infinitely many solutions.
In case of a system with more than two equations the system has a solution only if all the lines concur at a point.
The graphical solution for the system $y=2x-3$  and  $3x-y=4$  is shown below.  The point of intersection of the two lines is (1,-1).
Hence the solution to the system is x =1 and y =-1.



Solving  System of equations in three variables.

The three common methods used for solving the system of equations in two variables, the substitution, the elimination and the graphical methods can also be used for solving a system of equations in three variables.
The other two methods used for solving a system of three variables are
Cramers Rule:
This method is useful in determining the existence of a unique solution to the system and finding it. The System of equations are considered as a matrix arrangements and the determinants formed using the columns of the matrix are used in finding the solution of the system.  As determinants exist only for square matrices, this method can be applied only when the number of equations in the system is equal to the number variables.

Matrix row reduction
Using matrix row reduction steps, the matrix representing the given system can be reduced to a simple equivalent matrix. The solution of the system can be either got direct from the simpler matrix or by doing some back substitutions.  This method is also useful in determining the solutions in the case of the system having many solutions and also shows clearly when the system is inconsistent and does not have a solution. This method is adopted by computer programs used to determine the solutions of systems of many variables.