What is the solution of x+2y+z = -1 -x+3y+2z = 2 -y-2z =1

Question

I need to solve the system of linear equations with the cramer method.

Answer 1

First work out a determinant, by writing just the coefficients of the variables in the order x, y, z: Δ=

|  1  2  1 |

| -1  3  2 | = 1(-6+2)-2(2-0)+1(1-0)=-4-4+1=-7.

|  0 -1 -2 |

Now a determinant for each variable by replacing the variable column with the constants for each line: Δ_x=

| -1  2  1 |

|  2  3  2 | = -1(-6+2)-2(-4-2)+1(-2-3)=4+12-5=11.

|  1 -1 -2 |

x=Δ_x/Δ=-11/7.

Δ_y=

|  1 -1  1 |

| -1  2  2 | = 1(-4-2)+1(2-0)+1(-1-0)=-6+2-1=-5.

|  0  1 -2 |

y=Δ_y/Δ=5/7.

Δ_z=

|  1  2 -1 |

| -1  3  2 | = 1(3+2)-2(-1-0)-1(1-0)=5+2-1=6.

|  0  -1  1 |

z=Δ_z/Δ=-6/7.

SOLUTION: x=-11/7, y=5/7, z=-6/7.