2x1+0x2+3x3=3 4x1-3x2+7x3=5
8x1-9x2+15x3=10
solve the system using Gaussian or Gauss-Jordan elimination

[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.