A waitress sold 11 ribeye steak dinners and 18 grilled salmon dinners, totaling 592.29 on a particular day. Another day she sold 16 rib eye steak dinners and 6 grilled salmon dinners, totaling 580.56. How much did each type of dinner cost?

Let the cost of 1 ribeye steak dinner = x

The cost of 1 salmon dinner = y

Then,

11x + 18y = 592.29 (1)

16x + 6y = 580.56 (2)

Multiplying the second equation by 3, we get,

48x + 18y = 1741.68 (3)

(3) - (1) gives

37x = 1149.39

x = 31.06

Substituting the value for x in (1), we get,

11 (31.06) + 18y = 592.29

341.66 + 18y = 592.29

18y = 250.63

y = 13.92

Hence, the cost of ribeye steak dinner = 31.06 and the cost of grilled salmon dinner = 13.92.