Solve   x + y = 9
        y + z = 7
        x – z = 2

That is a dependent system and has an infinite
number of solutions.  I'll show you why that is
true, but we start out b y the elimination method:

Begin by writing the equations neatly so that 
like terms and equal signs line up vertically:

	
x + y     = 9
    y + z = 7
x     – z = 2

The idea is to create from this system of
3 equations in the same 3 unknowns to a system 
of two equations in the same two unknowns down 
to just 1 equation in 1 unknown, which we can
solve for a number and substitute it back in
previous equations to get the other two
unknowns.

The first equation has only 2 unknowns x and
y.  We can get another equations in only those
same two unknowns by adding the 2nd and 3rd
equations, so the the z's will cancel out:

    y + z = 7
x     – z = 2
-------------
x + y     = 9

So now we have two equations in the same two
variables and no others.

x + y     = 9
x + y     = 9

But, strangely enough, they are the same equation, 
and so the original system is dependent and there are 
infinitely many solutions.

Solve that for y

y = 9 - x

and solve  y + z = 7 for z

z = 7 - y

Now to find one solution:

For instance pick x = 1, and substitute in y = 9 - x

y = 9 - 1
y = 8

Substitute that in z = 7 - y
  
z = 7 - 8
z = -1

So one of many solutions is (x,y,z) = (1,8,-1)

For another solution, I'll arbitrarily pick x = -3, and 
substitute that in y = 9 - x

y = 9 - (-3)
y = ( + 3
y = 12

Substitute that in z = 7 - y
  
z = 7 - 12
z = -5

So another of the many possible solutions is (x,y,z) = (-3,12,-5)

Choose any values for x you like and the use those two equations to 
find y and z and find as many solutions to this system as you like.
They'll always check in the original equations.

Edwin