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Aragorn the Great has boasted to his hordes of followers that many a notorious villain has fallen to his awesome sword: His total of 560 victims consists of evil sorcerers, trolls, and orcs. These he has slain with a total of 620 mighty thrusts of his sword; evil sorcerers and trolls each requiring two thrusts (to the chest) and orcs each requiring one thrust (to the neck). When asked about the number of trolls he has slain, he replies, "I, the mighty Aragorn, despise trolls five times as much as I despise evil sorcerers. Accordingly, five times as many trolls as evil sorcerers have fallen to \(\mathrm{m} \mathrm{y}\) sword!" How many of each type of villain has he slain?

Short Answer

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Aragorn has slain 10 evil sorcerers, 50 trolls, and 500 orcs.
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Step 1: Equation for Total Victims

We are given that Aragorn has slain 560 victims in total, which include evil sorcerers, trolls, and orcs. Let x be the number of evil sorcerers, y be the number of trolls, and z be the number of orcs. Then we have the following equation for the total victims: x + y + z = 560

Step 2: Equation for Total Sword Thrusts

We are given that Aragorn used a total of 620 sword thrusts to slay the evil sorcerers, trolls, and orcs. Evil sorcerers and trolls require two sword thrusts each, while orcs only require one. Thus, we can write another equation representing the total sword thrusts: 2x + 2y + z = 620

Step 3: Relationship between Evil Sorcerers and Trolls

Finally, we are given that Aragorn has slain five times as many trolls as evil sorcerers. This can be written as the following equation: y = 5x

Step 4: System of Equations

We now have a system of three linear equations with three unknowns: 1. x + y + z = 560 2. 2x + 2y + z = 620 3. y = 5x

Step 5: Solving the System of Equations

To solve this system of equations, we can use the substitution method. We can substitute the third equation into the first and second equations: 1. x + 5x + z = 560 2. 2x + 2(5x) + z = 620 Simplifying this gives us: 1. 6x + z = 560 2. 12x + z = 620 Now solve the first equation for z: z = 560 - 6x Now substitute this into the second equation: 12x + (560 - 6x) = 620 Simplify and solve for x: 6x = 60 x = 10 Now we can find the values for y and z using the values we found for x: y = 5x = 5(10) = 50 z = 560 - 6x = 560 - 60 = 500

Step 6: Final Solution

So, Aragorn has slain 10 evil sorcerers, 50 trolls, and 500 orcs.

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