[1] … … 2x₁ + 0x₂ + 3x₃ = 3
[2] … … 4x₁ - 3x₂ + 7x₃ = 5
[3] … … 8x₁ - 9x₂ + 15x₃ = 10
Since [1] is already free of x₂, you might as well start by eliminating x₂ from the other two equations. Add -3 times [2] to [3] :
-3 (4x₁ - 3x₂ + 7x₃) + (8x₁ - 9x₂ + 15x₃) = -3 (5) + 10
-12x₁ + 9x₂ - 21x₃ + 8x₁ - 9x₂ + 15x₃ = -15 + 10
-4x₁ - 6x₃ = -5
[4] … … 4x₁ + 6x₃ = 5
Now eliminate x₃ in [1] and [4]. Add -2 times [1] to [4] :
-2 (2x₁ + 3x₃) + (4x₁ + 6x₃) = -2 (3) + 5
-4x₁ - 6x₃ + 4x₁ + 6x₃ = -6 + 5
0 = -1
which is a contradiction. This means the system has no solution.
