# Solve the following equations by method of reduction : x + 2y – z = 3 , 3x – y + 2z = 1 and 2x – 3y + 3z = 2 - Mathematics and Statistics

Sum

Solve the following equations by method of reduction :

x + 2y – z = 3 , 3x – y + 2z = 1 and 2x – 3y + 3z = 2

#### Solution

Matrix form of the given system of equations is

[(1, 2, 1),(3, -1, 2),(2, -3, 3)] [(x),(y),(z)] = [(3),(1),(2)]

This is of the form AX = B, where

A = [(1, 2, 1),(3, -1, 2),(2, -3, 3)], "X" =  [(x),(y),(z)] "and B"= [(3),(1),(2)]

Applying R2 → R2 – 3R1 and R3 → R3 – 2R1, we get

[(1, 2, 1),(0, -7, -1),(0, -7, 1)] [(x),(y),(z)] = [(3),(-8),(-4)]

Applying R3 → R3 – R2, we get

[(1, 2, 1),(0, -7, -1),(0, 0, 2)] [(x),(y),(z)] = [(3),(-8),(4)]

Hence, the original matrix A is reduced to an upper triangular matrix.

∴ [(x + 2y + z),(0 - 7y - zx),(0 + 0 + 2z)] = [(3),(-8),(4)]

By equality of martices, we get
x + 2y + z = 3         ...(i)
– 7y –z = – 8
i.e., 7y + z = 8        ...(ii)
2z = 4
∴ z = 2
Substituting z = 2 in equation (ii), we get
7y + 2 = 8
∴ 7y = 6
∴ y = (6)/(7)
Substituting y = 6)/(7) and z = 2 in equation (i), we get

x + 2(6/7) + 2 = 3

∴ x + 12/7 + 2 = 3

∴ x = 3 - 2 - 12/7 = (-5)/(7)

∴ x = (-5)/(7), y = (6)/(7) and z = 2 is the required solution.

Concept: Application of Matrices
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