Answer
Infinitely many solutions.
Dependent equations.
Work Step by Step
The given equations are
$\Rightarrow x+2y+z=4$ ......(1)
$\Rightarrow 3x-4y+z=4$ ......(2)
$\Rightarrow 6x-8y+2z=8$ ......(3)
Multiply equation (1) by $2$.
$\Rightarrow 2x+4y+2z=8$ ......(4)
Now add equation (2) and (4).
$\Rightarrow 3x-4y+z+2x+4y+2z=4+8$
Add like terms.
$\Rightarrow 5x+3z=12$ ...... (4)
Multiply equation (1) by $4$.
$\Rightarrow 4x+8y+4z=16$ ......(6)
Now add equation (3) and (6).
$\Rightarrow 6x-8y+2z+4x+8y+4z=8+16$
Add like terms.
$\Rightarrow 10x+6z=24$ ...... (7)
Multiply equation (4) by $-2$.
$\Rightarrow -10x-6z=-24$ ...... (8)
Add equation (7) and (8).
$\Rightarrow 10x+6z-10x-6z=24-24$
Add like terms.
$\Rightarrow 0=0$
Hence, the given system of equations is a dependent system.
There are infinitely many solutions.
