We can set up a system of equations to represent the number of each type of unit Dan created. Let x be the number of long swordsmen, y be the number of spearmen, and z be the number of crossbowmen. We know that the total number of units of each resource used to create the units must equal the amount of that resource that Dan has. We can write the following system of equations to represent this:
20x + 35y + 40z = 1375 (1)
10x + 40y + 30z = 2400 (2)
30z + 40z = 1300 (3)
Equation (1) represents the total number of units of food used to create the units, equation (2) represents the total number of units of wood used, and equation (3) represents the total number of units of gold used.
To solve this system of equations, we can use substitution. First, we'll solve for z in equation (3) by dividing both sides by 70:
z = 1300 / 70
z = 18.57
Since z represents the number of crossbowmen, it must be an integer. Since 18 is the largest integer less than 18.57, we know that Dan created 18 crossbowmen.
Now we can substitute 18 for z in equation (1) to find the number of long swordsmen Dan created:
20x + 35y + 40(18) = 1375
20x + 35y + 720 = 1375
20x + 35y = 655
We can divide both sides of this equation by 55 to solve for y:
(20x + 35y) / 55 = 655 / 55
y = 19
Since y represents the number of spearmen, it must be an integer. Since 19 is the largest integer less than 19.57, we know that Dan created 19 spearmen.
Now we can substitute 18 for z and 19 for y in equation (2) to find the number of long swordsmen Dan created:
10x + 40(19) + 30(18) = 2400
10x + 760 + 540 = 2400
10x = 1200
x = 120
Since x represents the number of long swordsmen, it must be an integer. Therefore, we know that Dan created 120 long swordsmen.
In total, Dan created 120 long swordsmen, 19 spearmen, and 18 crossbowmen.