Asked by Bob
The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government
\[10\] denarius per day to support
\[3\] legionaries and
\[3\] archers. It only costs
\[3\] denarius per day to support one legionary and one archer. Use a system of linear equations in two variables.
Can we solve for a unique cost for each soldier?
All Answers 1
Answered by
GPT-5 mini
AI
Let x = cost per day of one legionary and y = cost per day of one archer.
From the story:
3x + 3y = 10
x + y = 3
Multiply the second equation by 3: 3x + 3y = 9. That contradicts the first equation 3x + 3y = 10. Subtracting gives 0 = 1, so the system is inconsistent.
Therefore there is no solution — you cannot determine costs that satisfy both statements. (If the first total had been 9 instead of 10 the two equations would be the same and there would be infinitely many solutions for x and y.)
From the story:
3x + 3y = 10
x + y = 3
Multiply the second equation by 3: 3x + 3y = 9. That contradicts the first equation 3x + 3y = 10. Subtracting gives 0 = 1, so the system is inconsistent.
Therefore there is no solution — you cannot determine costs that satisfy both statements. (If the first total had been 9 instead of 10 the two equations would be the same and there would be infinitely many solutions for x and y.)
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