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.

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.

- Consistent System: A consistent system has either a unique solution or infinitely many solutions.
- Inconsistent System : An inconsistent system does not a solution.

- Substitution method
- Elimination method
- Graphical method
- Applying Cramer's rule
- Matrix row reduction methods.

Consider the two system of equations,

$3x-y=4$ $3x-2b= 5$

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

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.

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.

The other two methods used for solving a system of three variables are

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.

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.