Question

The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government 10 denarii per day to support 4 legionaries and 4 archers. It only costs 5 denarii per day to support 2 legionaries and 2 archers. Use a system of linear equations in two variables.
Can we solve for a unique cost for each soldier?

Choose 1 answer:

1) Yes; a legionary costs 11 denarius per day to support, and an archer costs 22 denarii per day to support.

2) Yes; a legionary costs 22 denarii per day to support, and an archer costs 4/3
denarii per day to support.

(3)
No; the system has many solutions.

(4)
No; the system has no solution.

Answer 1

Answer:

The 3rd option is the right option.

The system has Many solution.

Step-by-step explanation:

According to given data, we have two conditions.

⇒ It costs the Roman government 10 denarii per day to support 4 legionaries and 4 archers.

⇒ It only costs 5 denarii per day to support 2 legionaries and 2 archers.

Let the cost to support 1 Legionary = x

And the cost to support 1 Archer     = y

Then according to 1st condition

4 x + 4 y = 10

Dividing both sides by 2, we get

2 x + 2 y = 5  .................. (1)

And according to 2nd condition

2 x + 2 y = 5  .................. (2)

So basically, we have only ONE equation. And being a linear equation in 2 variables, it has many solutions.

Answer 2

3rd option The system has Many solution is right solution

we will explain it step by step

According to given data,

we have 2 conditions.

It costs the Roman government 10 denarii per day to support 4 legionaries and 4 archers.

It just costs 5 denarii per day to support 2 legionaries and 2 archers.

Let

the cost to support 1 Legionary = x

And

the cost to support 1 Archer     = y

according to first condition

4 x + 4 y = 10

we will Divide both sides by 2 then

we will get

2 x + 2 y = 5  .................. ...................................(a)

but according to second condition

2 x + 2 y = 5  ...........................................(b)

So we can clearly see tat we have only 1 equation. And being a linear equation in two variables,  it has many solutions.