find whether the following system of equations are consistent if so solve them x+2y+2z=2;3x-2y-z=5;2x-5y+3z=-4;x+4y+6z=0

Question

Matrices

Answer 1

x+2y+2z=2;

3x-2y-z=5;

2x-5y+3z=-4;

x+4y+6z=0

1  2   2 | 2

3 -2 -1  |5

2 -5  3  |-4

1  4   6 |0

On performing following operations we get Reduced Row Echelon Form

R2 = R2 - 3R1
R3 = R3 - 2R1
R4 = R4 - R1
R2 = R2/8
R1 = R1 - 2R2
R3 = R3 + 9R2
R4 = R4 - 2R2
R3 = (8/55)*R3
R1 = R1 - (1/4)R3
R2 = R2 - (7/8)R3
R4 = R4 - (9/4)R3

which is

1  0  0  |2

0  1  0  |1

0  0  1  |-1

0  0  0  |0

Hence the matrix is consistent.

And the solutions are:

x=2, y=1 and z = -1

Answer 2

The most straightforward way to check for consistency is to attempt to solve them. I will use basic algebra.

Label the equations:

A: x+2y+2z=2,

B: 3x-2y-z=5,

C: 2x-5y+3z=-4,

D: x+4y+6z=0.

From D, x=-4y-6z, so substitute for x in A and B (we can choose any two of A, B or C, so I chose A and B):

E: -4y-6z+2y+2z=-2y-4z=2,

F: -12y-18z-2y-z=5=-14y-19z=5.

Now we can solve for y and z.

-7E=14y+28z=-14.

-7E+F=28z-19z=-9, 9z=-9, z=-1.

So from E, -2y+4=2, 2y=2, y=1.

And from any of the equations A-D (for example, D):

x=-4+6=2.

So we have (x,y,z)=(2,1,-1) to check for consistency:

A: 2+2-2=2 OK

B: 6-2+1=5 OK

C: 4-5-3=-4 OK

D: 2+4-6=0 OK

So we have consistency x=2, y=1, z=-1.

Answer 3

X=2,y=1,z=-1