The system of equations may have a unique solution an infini

The system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the system, if a solution exists. (If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, express your answers in terms of z as in Example 3.)

Looking for answer in (x,y,z) form

Solution

for these equation matrix is

2 -5 1 -14

1 4 -6 -20

3 -4 -2 -28

by elementry row operation

1 0 -2 -12

0 1 -1 -2

0 0 0 0

by this matrix we get

x+0y -2z= -12

0x+y -z= -2

0x+0y+0z= 0

From these equation we get

x=-12+2z y=-2+z z=0

so (x,y,z) in z form is

(x,y,z)= (-12+2z,-2+z,0)