Contract curve for firms with linear utility functions

I am attempting to find a contract curve for a production economy with two linear utility functions.

Normally, I would find the point where the Marginal Rate of Technical Substitution were equal for the two firms. However, in this case, this does not seem to work (I end up getting an equation of two numbers).

How must I proceed?

Is it all places where both functions are equal?

• You have already asked this question, haven't you? – Giskard May 1 '16 at 9:32
• This is different from that. Both utility functions described for the production economy in this case are linear. – TheEconomist May 1 '16 at 9:37