The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government 101010 denarii per day to support 444 legionaries and 444 archers. It only costs 555 denarii per day to support 222 legionaries and 222 archers. Use a system of linear equations in two variables. Can we solve for a unique cost for each soldier?

There are many solutions

Let cost for each legionary be L

and that for each archer be A

We know that the system of linear equations that represent the information given in the question is:

444*L + 444*A = 101,010 222*L + 222*A = 555

We will notice that the above two equations represent the same line:

Divide the first equation by 222

⇒ 2*L + 2*A = 450

Now, divide the second equation by 11

⇒ 2*L + 2*A = 5

We can see that simplified versions of equations 1 and 2, lead to 2*L + 2*A being equal to different numbers

Hence, this system of equations is inconsistent. Hence, they have no solution, so we can't have a unique cost for each soldier.