Question

Express the following equations in matrix form and solve them by the method of reduction:

x − y + z = 1, 2x − y = 1, 3x + 3y − 4z = 2

Answer 1

The given equations can be written in the matrix form as:

`[(1,-1,1),(2,-1,0),(3,3,-4)] [("x"),("y"),("z")] = [(1),(1),(2)]`

By R₂ - 2R₁ and R₃ - 3R₁, we get,

`[(1,-1,1),(0,1,-2),(0,6,-7)] [("x"),("y"),("z")] = [(1),(-1),(-1)]`

By R₃ - 6R₂, we get,

`[(1,-1,1),(0,1,-2),(0,0,5)] [("x"),("y"),("z")] = [(1),(-1),(5)]`

∴ `[("x" - "y" + "z"),(0+"y" - 2"z"),(0 + 0 + "5z")] = [(1),(-1),(5)]`

By equality of matrices,

x - y + z = 1   ...(1)

y - 2z = - 1     ....(2)

5z = 5             ...(3)

From (3), z = 1

Substituting z = 1 in (2), we get,

y - 2 = - 1

∴ y = 1

Substituting y = 1, z = 1 in (1), we get,

x - 1 + 1 = 1

∴ x = 1

Hence, x = 1, y = 1, z = 1 is the required solution.