Question

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

`x + y = 1, y + z = 5/3, z + x 4/33`.

Answer 1

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

`[(1,1,0),(0,1,1),(1,0,1)] [("x"),("y"),("z")] = [(1),(5/3),(4/3)]`

By R₃ - R₁, we get,

`[(1,1,0),(0,1,1),(0,-1,1)] [("x"),("y"),("z")] = [(1),(5/3),(1/3)]`

By R₃ + R₂, we get,

`[(1,1,0),(0,1,1),(0,0,2)] [("x"),("y"),("z")] = [(1),(5/3),(2)]`

∴ `[("x" + "y" + 0),(0+"y" + "z"),(0 + 0 + 2"z")] = [(1),(5/3),(2)]`

By equality of matrices,

x + y = 1      ...(1)

y + z = `5/3`    .....(2)

2z = 2     ....(3)

From (3), z = 1

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

y + 1 = `5/3`

`y = 5/3 - 1`

∴ y = `2/3`

Substituting y = 2/3 in (1), we get,

∴ x + y = 1

`x + 2/3 = 1`

`x = 1 - 2/3 = 1/3`

∴ `x = 1/3`

Hence, x = `1/3`, y = `2/3`, z = 1 is the required solution.