A waitress sold $18$ ribeye steak dinners and $22$ grilled salmon dinners, totaling $\$ 579.15$ on a particular day. Another day she sold $29$ ribeye steak dinners and $11$ grilled salmon dinners, totaling $\$ 583.96$ . How much did each type of dinner cost? $\newline$ The cost of ribeye steak dinners is $\$ \square$ and the cost of salmon dinners is $\$ \square$ . $\newline$ (Simplify your answer. Round to the nearest hundredth as needed.)

Full solution

Q. A waitress sold

18

ribeye steak dinners and

22

grilled salmon dinners, totaling

\$ 579.15

on a particular day. Another day she sold

29

ribeye steak dinners and

11

grilled salmon dinners, totaling

\$ 583.96

. How much did each type of dinner cost?

\newline

The cost of ribeye steak dinners is

\$ \square

and the cost of salmon dinners is

\$ \square

\newline

(Simplify your answer. Round to the nearest hundredth as needed.)

Set up equations: Let $R$ be the cost of a ribeye steak dinner and $S$ be the cost of a grilled salmon dinner. We can set up two equations based on the information given: $18R + 22S = 579.15$ and $29R + 11S = 583.96$ .
Write first equation: Write the first equation from the given information: $18R + 22S = 579.15$ .
Write second equation: Write the second equation from the given information: $29R + 11S = 583.96$ .
Eliminate variable: Multiply the first equation by $11$ and the second equation by $22$ to eliminate $S$ when we subtract the equations. So we get $198R + 242S = 6370.65$ and $638R + 242S = 12847.12$ .
Solve for R: Subtract the second equation from the first to find $R$ : $198R + 242S - (638R + 242S) = 6370.65 - 12847.12$ . This simplifies to $-440R = -6476.47$ .
Plug in R: Divide both sides by $-440$ to solve for $R$ : $R = -6476.47 / -440$ . This gives us $R = 14.71925$ .
Solve for S: Now plug the value of $R$ back into the first equation to solve for $S$ : $18(14.71925) + 22S = 579.15$ . This simplifies to $264.9465 + 22S = 579.15$ .
Subtract values: Subtract $264.9465$ from both sides to solve for $S$ : $22S = 579.15 - 264.9465$ . This gives us $22S = 314.2035$ .
Find $S$ : Divide both sides by $22$ to find $S$ : $S = \frac{314.2035}{22}$ . This gives us $S = 14.28197727$ .
Round to nearest hundredth: Round $R$ and $S$ to the nearest hundredth: $R = \$14.72$ and $S = \$14.28$ .