Rahim goes to a grocery store and spends $\$ 22.00$ to purchase 3 cartons of strawberries and 2 cartons of raspberries. Paul goes to the same grocery store and spends $\$ 41.00$ to purchase 4 cartons of strawberries and 5 cartons of raspberries.
What is the price of one carton of strawberries and one carton of raspberries?

Step 1 ：Let the price of one carton of strawberries be s and the price of one carton of raspberries be r. We have the following system of linear equations:

Step 2 ：\(3s + 2r = 22\) (Rahim's purchase)

Step 3 ：\(4s + 5r = 41\) (Paul's purchase)

Step 4 ：Use the elimination method to solve the system of equations. Multiply the first equation by 2 to make the coefficients of r the same in both equations:

Step 5 ：\(6s + 4r = 44\)

Step 6 ：\(4s + 5r = 41\)

Step 7 ：Subtract the first equation from the second equation to eliminate r:

Step 8 ：\(-2s + r = -3\)

Step 9 ：Solve for r:

Step 10 ：\(r = 2s - 3\)

Step 11 ：Substitute this expression for r back into the first equation:

Step 12 ：\(3s + 2(2s - 3) = 22\)

Step 13 ：Solve for s:

Step 14 ：\(3s + 4s - 6 = 22\)

Step 15 ：\(7s = 28\)

Step 16 ：\(s = 4\)

Step 17 ：Find the value of r:

Step 18 ：\(r = 2(4) - 3\)

Step 19 ：\(r = 8 - 3\)

Step 20 ：\(r = 5\)

Step 21 ：Final Answer: The price of one carton of strawberries is \(\boxed{4}\) dollars and the price of one carton of raspberries is \(\boxed{5}\) dollars.