Hi. I'm trying to understand some functions of production from an economics book by Piero Sraffa. Since this is an algebraic question, I have not put it into "business math". Hope I did well.

On page 3 Sraffa writes:

Next, he writes: «There is a unique set of exchange values which if adopted by the market restores the original distribution of the product and makes it possible for the process to be repeated. This set is: 1 ton of iron for 10 quarters of wheat.»280 quarters of wheat and 12 tons of iron are used to produce 400 qr. of wheat. 120 quarters of wheat and 8 tons of iron are used to produce 20 tons of iron.

280 qr. wheat + 12 t iron --> 400 qr. wheat

120 qr. wheat + 8 t iron --> 20 t iron

It was quite easy for me to get this result by myself.

280 w + 12 i = 400 w

120 w + 8 t = 20 t

then

12 i = 120 w => 1 i = 10 w

After that, he makes an example with 3 goods, adding pigs

He says that we have the next functions of production:

And he adds that the only exchange value set in this case is the following:240 qr. wheat + 12 t of iron + 18 pigs --> 450 qr. wheat

90 qr. wheat + 6 t of iron + 12 pigs --> 21 tons of iron

120 qr. wheat + 3 t of iron + 30 pigs --> 60 pigs

10 qr. wheat = 1 t iron = 2 pigs

Now... I'm trying to get the same result by using a system of linear equations, as in the 2 goods model, but I can't get this result. Should I change my system to solve the equations? Or is the linear system a good solution and maybe I'm just a little rusty with algebra and calculations?

Of course I'm not asking for the solution, I'm just asking if, in your opinion, it's a good idea to use a system like this:

Thank you in advance.