help

Let's arrange them under each other 

x  +  y + 0z     = 3   (0*z will equal to 0 no matter what the value of z is, I written it to help visualizing) 

0x   + y +z       = 5

 x  +   0y +z     = 4

Now add these 3 equations

in the first column we have 2x, second column 2y and third column 2z and the last column the sum of the three numbers 3+5+4=12

So written as: 

2x+2y+2z=12

Factor 2

2(x+y+z)=12

divide by 2

x+y+z=12/2=6

x+y+z=6 

If x + y = 3, y + z = 5, and z + x = 4, then find x + y + z.

x +  y  =   3        (1)

y +  z  =   5        (2)

x + z  =    4        (3)

Subtract   (3)  from (2)   and we get that

y - x  =  1    ⇒    -x  + y  =   1     (4)

Add (1)  and (4)

2y  =  4

y  = 2

And   x  + y  = 3    implies that x  = 1

And  x + z =  4     implies that z  = 3

So

x + y +  z  =   1   + 2  +   3    =   6