At a ski resort, one family pays \( \$ 1,045 \) for three nights of lodging and 8 ski lift passes. Another family pays \( \$ 1,305 \) for three nights of lodging and twelve ski lift passes. What is the cost of one night's lodging and what is the cost of one ski lift pass?

Let's denote the cost of one night's lodging as \( L \) and the cost of one ski lift pass as \( P \). From the information provided, we can set up the following two equations based on the costs for the two families: 1. For the first family: \( 3L + 8P = 1045 \) (Equation 1) 2. For the second family: \( 3L + 12P = 1305 \) (Equation 2) Now, we can solve these equations step-by-step. First, subtract Equation 1 from Equation 2: \[ (3L + 12P) - (3L + 8P) = 1305 - 1045 \] This simplifies to: \[ 4P = 260 \] Dividing both sides by 4 gives: \[ P = 65 \] Now that we have the cost of one ski lift pass, we can substitute \( P \) back into Equation 1 to find \( L \): \[ 3L + 8(65) = 1045 \] This simplifies to: \[ 3L + 520 = 1045 \] Subtract 520 from both sides: \[ 3L = 525 \] Now, divide both sides by 3: \[ L = 175 \] Thus, the cost of one night's lodging is \( \$175 \), and the cost of one ski lift pass is \( \$65 \).