A neighborhood group goes to the movies every Saturday night. Last week, they purchased 29 adult tickets and 17 youth tickets for $\$ 254$. This week they purchased 21 adult tickets and 22 youth tickets for $\$ 213$. Find the cost of one adult ticket. $\$ 7$ $\$ 3$ $\$ 9$ $\$ 6$

Step 1 ：Let's denote the cost of one adult ticket as 'a' and the cost of one youth ticket as 'y'.

Step 2 ：From the first week's purchase, we can form the equation \(29a + 17y = 254\).

Step 3 ：From the second week's purchase, we can form the equation \(21a + 22y = 213\).

Step 4 ：These form a system of linear equations that we can solve to find the values of 'a' and 'y'.

Step 5 ：Solving this system, we find that 'a' equals 7 and 'y' equals 3.

Step 6 ：Thus, the cost of one adult ticket is \(\boxed{7}\) dollars.