System of Linear Non Homogenous Simultaneous Equations

System of Linear Non Homogenous Simultaneous Equations

Consider the system of linear non–homogenous simultaneous equations in three unknowns x, y and z, given by a₁x + b₁y + c₁z = d₁, a₂x + b2y + c₂z = d₂ and a₃x + b₃y + c₃z = d₃.

where (d₁, d₂, d₃) ≠ (0, 0, 0).

Let

obtained on replacing first column of Δ by B.

Similarly

It can be shown that AX = B, x.Δ = Δx, y.Δ = Δy, z.Δ = Δz.

(a) Determinant Method of Solution

(i) If Δ ≠ 0, then the given system of equations has unique solution, given by

x = Δx/Δ, y = Δy/Δ and z = Δz /Δ.

(ii) If Δ = 0, then two sub cases arise:

(α) at least one of Δx, Δy and Δz is non–zero, say Δx ≠ 0. Now in x.Δ = Δx, L.H.S. is zero and R.H.S. is not equal to zero. Thus we have no value of x satisfying

x.Δ = Δx. Hence given system of equations has no solution.

(β) Δx = Δy = Δz = 0. In the case the given equations are dependent. Delete one or two equation from the given system (as the case may be) to obtain independent equation(s). The remaining equation(s) may have no solution or infinitely many solutions. For example in x + y + z = 3, 2x + 2y + 2z = 9, 3x + 3y + 3z = 12,
Δ = Δx = Δy = Δz = 0 and hence equations are dependent (infact third equation is the sum of first two equations). Now after deleting the third equation we obtain independent equations x + y + z = 3, 2x + 2y + 2z = 9, which obviously have no solution (infact these are parallel planes) where as in x + y + z = 3,
2x – y + 3z = 4, 3x + 4z = 7, Δ = Δx = Δy = Δz = 0 and hence equations are dependent (infact third equation is the sum of first two equations). Now after deleting any equation (say third) we obtain independent equations
x + y + z = 3, 2x – y + 3z = 4, which have infinitely many solutions (infact these are non parallel planes). For let z = k ∈ R, then and . Hence we get infinitely many solutions.

(b) Matrix Method of Solution

(i) If Δ ≠ 0, then A–1 exists and hence AX = B ⇒ A–1(AX) = A–1B ⇒ X = A–1B and therefore unique values of x, y and z are obtained.

(ii) Ìf Δ = 0, then we from the matrix [A : B], known as augmented matrix (a matrix of order 3 × 4). Using row operations obtain a matrix from [A : B] whose last row corresponding to A is zero (this is possible as |A| = 0). If last entry of B in this matrix is non–zero, then the system has no solution else the given equations are dependent. Proceed further in the same way as in the case of determinant method of solution discussed earlier.