Two Variable Problem

I need some help with a timing scenario. I have a belted
conveyor which carries a product toward a flighted machine.
At the end of the belted conveyor is a device which can lift
the product above the belts, stopping it until a proper time
to release it in time with the flighted machine. I receive
one pulse per flight from the machine. Up until now the
speed of my belted conveyor was constant and that of the
flighted machine was variable.

I took the time between the pulses from the flighted machine
and used it as 'x' in the old 'y = mx + b' formula. It wasn't
hard deriving the 'm' and the 'b' by two settings at a high
speed and low speed of the flighted machine. For this
'one-variable' problem it was easy applying the slope and
y-intercept method we all learned in algebra. This gave me
a delay time after the pulse at which to release the product.
Worked great. Now here's the problem.

They want to make the belted conveyor's speed variable also.
I know I'll have a formula like 'z = ax + by + c' with 'z' as
my resultant timing, 'a' as the multiplier times the timing
from the flighted machine, 'b' as the multiplier times a similar
timing I'll get from my, now variable, belted conveyor, and 'c'
as an offset. What do I do to derive the 'a', 'b' and
'c' in my new scenario? I'm sorry for the length of the question
but I think it's important to understand the setup.

If you have a table of x,y, and desired z..

I can calculate the a and b constants using a least squares technique.

I am working now, but it is easy with Mathcad.

You might get by like this:

assume x is 0. Now find the soluition to z = b*y +c

next assume y is 0. Now find the solution to z = a*x+c

The least squares can solve for x,y, z at once.

Thanks Peter for your quick reply. It has jogged some brain cells. After my lunch and a couple of games of Freecell (to get the mind off the problem for a moment) I think I'm now coming up with a method that I can do on the PLC (AB MicroLogix 1200 by the way).

Try Freecell number 1942

I also made an error in my last post. It should say solve for a,b and c at the same time.