A waitress sold $18$ ribeye steak dinners and $12$ grilled salmon dinners, totaling $\$ 563.44$ on a particular day. Another day she sold $26$ ribeye steak dinners and $6$ grilled salmon dinners, totaling $\$ 580.72$ . How much did each type of dinner cost? $\newline$ The cost of ribeye steak dinners is $\$$ $\square$ and the cost of salmon dinners is $\$$ $\square$ . (Simplify your answer. Round to the nearest hundredth as needed.)

Full solution

Q. A waitress sold

18

ribeye steak dinners and

12

grilled salmon dinners, totaling

\$ 563.44

on a particular day. Another day she sold

26

ribeye steak dinners and

6

grilled salmon dinners, totaling

\$ 580.72

. 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

. (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. Equation $1$ : $18R + 12S = 563.44$ . Equation $2$ : $26R + 6S = 580.72$ .
Modify second equation: Multiply the second equation by $2$ to make the coefficient of $S$ the same as in the first equation. So, $2 \times (26R + 6S) = 2 \times 580.72$ , which gives us $52R + 12S = 1161.44$ .
Eliminate S: Now subtract the first equation from the modified second equation to eliminate S. 52R + 12S) - (18R + 12S) = 1161.44 - 563.44\. This simplifies to \$34R = 598.
Solve for R: Divide both sides of the equation by $34$ to solve for $R$ . $\frac{34R}{34} = \frac{598}{34}$ , which gives us $R = 17.59$ .
Plug in R: Now plug the value of R back into one of the original equations to solve for S. Using the first equation: $18(17.59) + 12S = 563.44$ . This simplifies to $316.62 + 12S = 563.44$ .
Solve for S: Subtract $316.62$ from both sides to solve for $S$ . $12S = 563.44 - 316.62$ , which gives us $12S = 246.82$ .
Final solution: Divide both sides by $12$ to find the value of $S$ . $\frac{12S}{12} = \frac{246.82}{12}$ , which gives us $S = 20.57$ .