System of linear equation

 

Linear Equations

The equation of the form ax + b = 0, is called the equation of first degree in x. Such equation gives a single value of x satisfying it. This value of the value of x satisfying the equation is called the root of the equation or the solution of the equation.

For example, 2x - 4 = 0 is satisfied by x = 2, so x = 2 is the root or solution of the equation.

Similarly, the equation of the form ax + by = c is called the equation of first degree in x and y. Infinite number of the values of x and y satisfy this equation. If we plot these points (x, y) in the graph, it gives a straight line. For example,

2x + 3y = 6 is satisfied by (3, 0), (0, 2), (6, 2), (- 3, 4), etc. These points lie in a straight lines.

    System of Linear Equation

The pair of linear equations of the form

            a₁x + b₁y = c₁

            a₂x + b₂y = c₂

is referred as the system of the equations. The typical example of system of linear equations are

            i) 2x + 3y = 6                    ii) x - 2y = 3                  iii) x + 4y = 10

               x - y = 2                            2x - 4y = 2                       x + 4y = 12

In above examples, both the equations of (i) are satisfied by x = 4, y = 2. The solution of this equation is unique. Such system of equations which has a solution and the solution is unique, the system of equations is said to be consistent and independent. The straight lines of such equations in graph intersect at a point and this point is the solution of the system of equations.

 

For the system of equations (ii), infinite number of values of x and y such as (5, 1), (7, 2), (9, 3) etc. satisfy it. The equations which has infinite number of solutions satisfying them, the system of equations are called consistent and dependent. The straight lines of such equations in graph are coincide or are overlap to each other.

For the equations in (iii), no values of x and y satisfy these equations. Such system of equations which has no solutions satisfying them are called inconsistent and independent. The straight lines in graph are parallel to each other.

Similarly, the equation of the form

            ax + by + cz = d is called general equation of first degree in x, y and z. The solution of this equations consists of the order triples. The particular example of such equation is

            2x + 3y + 4z = 12, one of the solution of this equation is the order triple (1, 2, 1).

Three equations

            a₁x + b₁y + c₁z = d₁

            a₂x + b₂y + c₂z = d₂

            a₃x + b₃y + c₃z = d₃

are referred as the system of linear equations in x, y and z. As in equation of two variable, the system of equations of three variables may have unique solution (consistent and independent); it may have infinite solution (consistent and dependent) or it may not have solution satisfying it                 ( inconsistent and independent).

Solution of System of Linear Equations

There are different method for solving the system of linear equations. Some of the method of solving the liner equations are; method of substitution, eliminating method, row equivalent matrix method, Cramer's rule, inverse matrix method, graphical method etc. In this chapter, we discuss only three methods of solving system of linear equations; (a) row equivalent matrix method (b) Cramer's Rule and (c) matrix inverse methods.