What can be said about 3 variables if only 2 equations are given?

If we are given 2 equations with three variables, they can
be of the form

a1x+ b1y+ c1z+d1=0…
(1)

a2x+ b2y+ c2z+d2=0…
(2)

Now as we have 3 variables we can only eliminate one of
the variables using the two equations, so we have to express two of the variables as
factors of the third variable.

For example, if we perform
a2*(1) - a1*(2)

We can eliminate x we are left with a2b1y+
a2c1z+a2d1- a1b2y-a1c2z-a1d2=0

=> y= [a1d2- a2d1
+a1c2z – a2c1z]/ (a2b1- a1b2)

Similarly, x also can be
written as an expression with z in it. It is not possible to find numeric values of x, y
and z if only two equations are given.