## System of Simultaneous Linear Equations

System of Simultaneous Linear EquationsConsider the following system of n linear equations in n unknowns:

a

_{11}x_{1}+ a_{12}x_{2}+.........+ a_{1n}x_{n}= d_{1}a

_{21}x_{1}+ a_{22}x_{2}+.........+ a_{2n}x_{n}= d_{2}. . . .

. . . .

a

_{n1}x_{1}+ a_{n2}x_{2}+.........+ a_{nn}x_{n}= d_{n}This system of equation can be written in the matrix form as

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 EquationsA 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 x

_{1}, x_{2}, ......, x_{n}which simultaneously satisfy all the equations is called a solution of the system of equations.

Consistent SystemIf 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

a

_{1}x + b_{1}y + c_{1}z = d_{1},a

_{2}x + b_{2}y + c_{2}z = d_{2}, ...... (i)a

_{3}x + b_{3}y + c_{3}z = d_{3}.If A = X = and D =

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 getA

^{-1}(AX) = A^{-1}D => IX = A^{-1}D [·.· A^{-1}A = I]=> X = A

^{-1}D => ...... (iii)where A

_{1}, B_{1}etc. are the cofactors of a_{1}, b_{2}etc. in the determinantΔ = (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 C

_{2}→ C_{2}- aC_{1}, C_{3}→ C_{3}+ C_{1}, we getA = = 0.

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

^{2}) = 0=> (a + 2) (a

^{2}- 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)

=> k

^{2}- 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.