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.