Awaitress sold 17 ribeye steak dinners and 16 grilled salmon dinners, totaling $571.16 on a particular day. another day she sold 21 ribeye steak dinners and 8 grilled salmon dinners, totaling $583.22. how much did each type of dinner cost?

Let x = ribeye steak cost, y = grilled salmon cost. With this we set up two equations:

17x + 16y = 571.16
21x + 8y = 583.22

We must now pick which dinner we could like to find the cost of first. I choose the ribeyes. To do so, I'm going to divide the first equation by - 2. I will explain why:

- 8.5x - 8y = - 285.58
21x + 8y = 571.16

Now we combine like terms. Notice that the y-variables cancel each other out. This why I divided by - 2. It causes this cancellation leaving us with only one variable to solve. This gives us:

12.5x = 297.64

Divide both sides by 12.5 to isolate variable x.

x = 23.8112 or x ≈ 23.81. I round two decimal points since we're dealing with money. 

Now, we plug this cost into either of the two equations and solve for variable y. I'll use the first one.

17(23.81) + 16y = 571.16
404.77 + 16y = 571.16

Subtract 404.77 from both sides to isolate the variable and its coefficient. 

16y = 166.37

Divide by 16 on both sides to get rid of the coefficient

y ≈ 10.3993 or y ≈ 10.40 since we're dealing with money. 

Cost of ribeye steak -- $23.81
Cost of grilled salmon -- $10.40