Step 1 :Define two variables, s for the number of sliders and o for the number of onion rings.
Step 2 :Write the first equation to represent the total number of sliders and onion rings: \(s + o = 11\).
Step 3 :Write the second equation to represent the total number of calories: \(350s + 40o = 1370\).
Step 4 :Solve the system of equations to find the values of s and o.
Step 5 :Final Answer: The system of equations is \(\boxed{s + o = 11}\) and \(\boxed{350s + 40o = 1370}\), where s represents the number of sliders and o represents the number of onion rings.