# Solve the following equations by method of reduction : x – 3y + z = 2 , 3x + y + z = 1 and 5x + y + 3z = 3 - Mathematics and Statistics

Sum

Solve the following equations by method of reduction :

x – 3y + z = 2 , 3x + y + z = 1 and 5x + y + 3z = 3

#### Solution

Matrix form of the given system of equations is

[(1, -3, 1),(3, 1, 1),(5, 1, 3)] [(x),(y),(z)] = [(2),(1),(3)]

This is of the form AX = B, where

A = [(1, -3, 1),(3, 1, 1),(5, 1, 3)], "X" = [(x),(y),(z)] "and B"= [(2),(1),(3)]

Applying R2 → R2 – 3R1 and R3 → R3 – 5R1, we get

[(1,  -3, 1),(0, 10, -2),(0, 16, -2)] [(x),(y),(z)] = [(2),(-5),(-7)]

Applying R3 → R3 – (8/5) R2, we get

[(1, -3, 1),(0, 10, -2),(0, 0, 6/5)][(x),(y),(z)] = [(2),(-5),(1)]

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

∴ [(x - 3y + z),(0 + 10y - 2z),(0 + 0 + 6/5z)] = [(2),(-5),(1)]

∴ By equality of martices, we get

x – 3y + z = 2        ...(i)

10y – 2z = – 5           ...(ii)

(6)/(5)z = 1

∴ z = (5)/(6)

Substituting z = (5)/(6) in equation (ii), we get

10y - 2(5/6) = – 5

∴ 10y - (10)/(6) = – 5

∴ 10y = -5 + (10)/(6) = (-20)/(6)

∴ 10y = (-10)/(3)

∴ y = (-1)/(3)

Substituting y = (-1)/(3) and z = (5)/(6) in equation (i), we get

x - 3((-1)/3) + (5)/(6) = 2

∴ x + 1 + (5)/(6) = 2

∴ x = 2 - 1 - (5)/(6) = (1)/(6)

∴ x = (1)/(6), y = (-1)/(3) and z = (5)/(6) is the required solution.

Concept: Application of Matrices
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