Let's denote the cost of the presents with the following variables:
- \( R \) for the cost of the red present
- \( P \) for the cost of the purple present
- \( B \) for the cost of the blue present
- \( G \) for the cost of the gold present
- \( N \) for the cost of the green present
From the information given, we can form the following equations:
- \( R + P = 38 \) (Equation 1)
- \( P + B = 31 \) (Equation 2)
- \( B + G = 25 \) (Equation 3)
- \( G + N = 23 \) (Equation 4)
- \( N + R = 23 \) (Equation 5)
Now, we'll solve these equations step by step.
Step 1: Solve for \( R \) in terms of \( P \) using Equation 1: \[ R = 38 - P \]
Step 2: Solve for \( P \) in terms of \( B \) using Equation 2: \[ P = 31 - B \]
Step 3: Substitute the expression for \( P \) into the expression for \( R \): \[ R = 38 - (31 - B) = 38 - 31 + B = 7 + B \]
Step 4: Solve for \( B \) in terms of \( G \) using Equation 3: \[ B = 25 - G \]
Step 5: Substitute \( B = 25 - G \) into \( R = 7 + B \): \[ R = 7 + (25 - G) = 32 - G \]
Step 6: Now solve for \( G \) in terms of \( N \) using Equation 4: \[ G = 23 - N \]
Step 7: Substitute \( G = 23 - N \) into \( R = 32 - G \): \[ R = 32 - (23 - N) = 32 - 23 + N = 9 + N \]
Step 8: Solve for \( N \) in terms of \( R \) using Equation 5: \[ N = 23 - R \]
Step 9: Now substitute \( R = 9 + N \) into \( N = 23 - R \): \[ N = 23 - (9 + N) \ N + N = 23 - 9 \ 2N = 14 \ N = 7 \]
Step 10: Now substitute \( N = 7 \) back to find \( R \): \[ R = 9 + 7 = 16 \]
Step 11: Now find \( G \) using \( G = 23 - N \): \[ G = 23 - 7 = 16 \]
Step 12: Find \( B \) using \( B = 25 - G \): \[ B = 25 - 16 = 9 \]
Step 13: Find \( P \) using \( P = 31 - B \): \[ P = 31 - 9 = 22 \]
Step 14: Finally, with all present costs calculated, we summarize as follows:
- \( R = 16 \)
- \( P = 22 \)
- \( B = 9 \)
- \( G = 16 \)
- \( N = 7 \)
To find the total cost of all presents, we sum these amounts: \[ R + P + B + G + N = 16 + 22 + 9 + 16 + 7 = 70 \]
Thus, the total cost of all five presents is \( \boxed{70} \).