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System of Simultaneous Linear Equations

System of Simultaneous Linear Equations

Consider the following system of n linear equations in n unknowns:

a11 x1 + a12 x2 +.........+ a1n xn = d1

a21 x1 + a22 x2 +.........+ a2n xn = d2        .                              .   .

an1 x1 + an2 x2 +.........+ ann xn = dn

This system of equation can be written in the matrix form as

system-of-equation

or AX = D.

The n × n matrix A is called the coefficient matrix of the system of linear equations.

Homogeneous and Non-Homogeneous System of Linear Equations

A system of equations AX = D is called a homogeneous system if D = O. Otherwise it is called a non-homogeneous systems of equations.

Solution of a System of Equations

Consider the system of equation AX = D.

A set of values of the variables x1, x2, ......, xn which simultaneously satisfy all the equations is called a solution of the system of equations.

Consistent System

If the system of equations has one or more solutions, then it is said to be a consistent system of equations, otherwise it is an inconsistent system of equations.

Solution of a Non-Homogeneous System of Linear Equations

  There are two methods of solving a non-homogeneous system of simultaneous linear equations.

(i)    Cramer's Rule

(ii)    Matrix Method

(i)     Cramer's Rule:

It is discussed under the topic of Determinants.

(ii)    Matrix Method:

Consider the equations

a1x + b1y + c1z = d1,

a2x + b2y + c2z = d2,                                                           ...... (i)

a3x + b3y + c3z = d3.

If A = matrix31     X = matrix32       and D = matrix33

then the equation (i) is equivalent to the matrix equation

A X = D.                                                                            ...... (ii)

Multiplying both sides of (ii) by the inverse matrix A-1, we get

A-1 (AX) = A-1 D => IX = A-1D         [·.· A-1 A = I]

=> X = A-1 D => matrix34                                  ...... (iii)

where A1, B1 etc. are the cofactors of a1, b2 etc. in the determinant

Δ = matrix35       (Δ ≠ 0).

(i)     If A is a non-singular matrix, then the system of equations given by
AX = D has a unique solutions given by X = A-1 D.

(ii)    If A is a singular matrix, and (adjA)D = O, then the system of equations given by AX = D is consistent, with infinitely many solutions.

(iii)    If A is a singular matrix, and (adjA)D ≠ O, then the system of equation given by AX = D is inconsistent.

Solution of Homogeneous System of Linear Equations:

Let AX = O be a homogeneous system of n linear equation with n unknowns. Now if A is non-singular then the system of equations will have a unique solution i.e. trivial solution and if A is singular then the system of equations will have infinitely many solutions.

Illustration:

If the system of equations x + ay - z = 0, 2x - y + az = 0 and
ax + y + 2z = 0 has a non trivial solution, then find the value of 'a'.

Solution:

        Using C2 → C2 - aC1, C3 → C3 + C1, we get

        A = matrix36 = 0.

        => (2 + a)(-1 -2a - 1 + a2) = 0

        => (a + 2) (a2 - 2a - 2) = 0

        => a = -2, a = 1 + √3.

Illustration:

Find the value of 'k' for which the system of equations (k + 1)
x + 8y = 4k, kx + (k + 3)y = 3k -1 has no solution.

Solution:

        For the system of equations to have no solution, we must have

        (k+1)/k = 8/(k+3) ≠ 4k/(3k-1)

        => (k + 1) (k + 3) = 8k and 8 (3k - 1) ≠ 4k (k + 3)

        => k2 - 4k + 3 = 0 => k = 1, 3.

        For  = 1, 8(3k - 1) = 16 and 4k (k + 3) = 16.

        For k = 3, 8(3k - 1) = 64 and 4k (k + 3) = 72.

        .·. for k = 3, 8(3k - 1) ≠ 4k (k + 3).

        .·. k = 3 is the required value of 'k' for no solution.

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