# 2x – y + 3z = 8, X – 6y – z = 0, -6x +3y – 9z =24

solve system of equation with 3 variables

2x-y+3z  = 8 -------> (1)

x-6y-z = 0 ---------> (2)

-6x+3y-9z = 24 ---------> (3)

Multiple to each side of equation (2) by 2.

2x-12y-2z = 0 ----------> (4)

To eliminate the x value subtract (4) from (1).

2x-12y-2z = 0

2x -y   +3z = 8

(-) (+) (-)     (-)

___________

-11y-5z = -8

11y+5z = 8 -----------> (5)

multiple to each side of equation (2) by 6.

6x-36y-6z = 0-------> (6)

To eliminate the x value add equations (3) and (6).

6x-36y-6z = 0

-6x+3y-9z =24

_______________

-33y-15z =24 -----------> (7)

Multiple to each side of equation (5) by 3.

33y+15z = 24 ----------> (8)

Observe the given equations (1) and (3) are parallel.

parallel lines does not intersect.

So given system has no soluition.

The equations are 2x - y + 3z = 8

x - 6y - z = 0

- 6x + 3y - 9z = 24

Form a system with the equations of the planes and calculate ranks.

The two rows of the coefficient matrix are proportional.

Two parallel planes and the other cuts each in a line.