Let's start by finding the cost per kilogram of each alloy.
For the first purchase:
- 48 kilograms of iron alloy and 40 kilograms of lead alloy cost $1224.
- Let x be the cost per kilogram of the iron alloy and y be the cost per kilogram of the lead alloy.
- We can write a system of two equations to represent the cost of the alloys:
```
48x + 40y = 1224
x + y = ?
```
We don't know the total cost per kilogram of both alloys combined, so we'll use a variable to represent it: `x + y = z`.
For the second purchase:
- 24 kilograms of iron alloy and 13 kilograms of lead alloy cost $507.
- We can use the total cost per kilogram of both alloys combined to find the cost of each alloy. We know that the total cost per kilogram is the same for both purchases, so we can write:
```
24x + 13y = 507
```
Now we can solve the system of equations:
```
48x + 40y = 1224
24x + 13y = 507
```
Multiplying the second equation by 2 to get the same number of x's as the first equation:
```
48x + 26y = 1014
48x + 40y = 1224
```
Subtracting the first equation from the second:
```
14y = 210
y = 15
```
Substituting y into the first equation:
```
48x + 40(15) = 1224
48x = 624
x = 13
```
So the cost per kilogram of the iron alloy is $13 and the cost per kilogram of the lead alloy is $15.
To find the cost of each alloy:
- For the first purchase, the cost of the 48 kilograms of iron alloy is 48 x $13 = $624, and the cost of the 40 kilograms of lead alloy is 40 x $15 = $600.
- For the second purchase, the cost of the 24 kilograms of iron alloy is 24 x $13 = $312, and the cost of the 13 kilograms of lead alloy is 13 x $15 = $195.
So the total cost of the iron alloy is $624 + $312 = $936, and the total cost of the lead alloy is
!