A 6000 -seat theater has tickets for sale at $\$ 28$ and $\$ 40$. How many tickets should be sold at each price for a sellout performance to generate a total revenue of $\$ 194,400$ ? The number of tickets for sale at $\$ 28$ should be

Step 1 ：Let's denote the number of tickets sold at $28 as x and the number of tickets sold at $40 as y. We know that the total number of tickets is 6000, so we have the equation \(x + y = 6000\). We also know that the total revenue is $194,400, so we have the equation \(28x + 40y = 194400\). We can solve this system of equations to find the values of x and y.

Step 2 ：The solution to the system of equations is \(x = 3800\) and \(y = 2200\). This means that 3800 tickets should be sold at $28 and 2200 tickets should be sold at $40 to generate a total revenue of $194,400.

Step 3 ：Final Answer: The number of tickets for sale at $28 should be \(\boxed{3800}\).