Mathematical Equation help?

Question

Guys I realy need help on these two equations..

The first one, we will have to solve by performing Row Operations

2x+2y+2z= 6
3x-3y-4z= -1
x+y+3z= 11

Can you guys please tell me whether it is Inconsistent , dependent, or independent please and why?

Please work it out for me I would SOO much appreciate it if you displayed how you came to this.

Answer 1

simultaneous equations

2x+2y+2z =6 1

x+y+3z =11 2

(2)*2

2x+2y+6z =22
(2)-(1)

4z=16

z = 4


substiture z in to (1)

2x+2y = -2
x+y = -1

substitue z=4 into the 2nd equation

3x-3y-16 =-1

3x-3y =15

x-y = 5


x= 2


therefore
y = -3

x=2,y= -3,z =4

Answer 2

You can divide both sides of equ.1 by 2 and substract it from the third
x+y+3z-11-( x+y+z-3) =0==>2z-8=0 z=4
now we have the system

x+y =-1

3x-3y =15

multiplying the first by 3 and summing
6x=12 x=2
from 2+y=-1 you get y=-3
Verifying

2*2-2*3+8=6
3*2+9 -16 =-1
2-3+12 =11
The system has one solution so the tthree equ.are independent.

I wonder if you know about determinants if yes,the determinant
of this system is

D= I1 1 1 I = -9 -4 +3 +3+12 +4 =9 not zero.
I3 -3 -4I
I1 1 3 I

That means that the equ.are independent.and you can tellso without solving the system

Answer 3

2x+2y+2z= 6 <-- EQ 1
3x-3y-4z= -1 <-- EQ 2
x+y+3z= 11 <-- EQ 3
Multiply EQ 3 by -2 getting:
-2x-2y-6z = -33 <-- EQ 4
Add EQ 4 to EQ 1 getting:
-4z = - 27
z = 27/4 = 6.75
Replace z in EQ 1 and EQ2 with 6.75 getting:
2x+2y + 13.5 = 6 so 2x+2y = - 7.5 <-- EQ 5
3x-3y - 31 = -1 so 3x-3y = 30 <-- EQ 6
Multiply Eq 5 by 3 and EQ 6 by 2 getting:
6x+6y = -22.5 <-- EQ 7
6x-6y= 180 <-- EQ 8
Add EQ 7 and EQ 8 getting:
12x = 157.5
x = 13.125
Replace x in EQ 8 with 13.125 getting
6(13.125) -6y = 180
-6y = 101.25
y = -16.875
The equations are independent because we found a solution.