Two families go to a hockey game. One family purchases two adult tickets and four youth tickets for \( \$ 33 \). Another family purchases four adult tickets and five youth tickets for \( \$ 51.75 \). What is the cost of an adult ticket and the cost of a youth ticket?

Step 1 ：1. \(2a + 4y = 33\)

Step 2 ：2. \(4a + 5y = 51.75\)

Step 3 ：3. \(a = \frac{51.75 - 5y}{4}\)

Step 4 ：4. Substitute \(a\) from step 3 into step 1: \(2\left(\frac{51.75 - 5y}{4}\right) + 4y = 33\)

Step 5 ：5. Solve for \(y\): \(y = 6.25\)

Step 6 ：6. Substitute \(y\) back into step 3 to find \(a\): \(a = \frac{51.75 - 5(6.25)}{4}\)

Step 7 ：7. Solve for \(a\): \(a = 3.5\)