explain to me exactly how to use Craimer's rule and the Row-Echelon form?

Question

Solve

2x - y - z = 15

4x + 5y + 2z = 10

-x - 4y + 3y = -20

Answer 1

The equations are 2x - y - z = 15

4x + 5y + 2z = 10

- x - 4y + 3y = - 20

Write the equations into matrices form AX  = B .

A is Coefficient matrix , X is variable matrix and B is Constant matrix.

Solve by Cramer's rule

Det A = 2(15 + 8) + 1(12 + 2) - 1(- 16 + 5)

= 46 + 14 + 11

= 71

Det A₁ = 15(15 + 8) + 1(30 + 40) - 1(- 40 + 100)

= 345 + 70 - 60

= 355

DetA₂ = 2(30 + 40)  - 15(12 + 2) - 1(- 80 + 10)

= 140 - 210 + 70

= 0

DetA3 = 2(- 100 + 40) + 1(- 80 + 10) + 15(- 16 + 5)

= - 120 - 70 - 165

= - 355

x = Det A₁ /DetA = 355/71 = 5

y = DetA₂/DetA = 0/71 = 0

z = DetA3/DetA = - 355/71 = -5

Solution x = 5, y = 0 and z = -5.