A waitress sold 14 ribeye steak dinners and 26 grilled salmon​ dinners, totaling ​$568.69 on a particular day. Another day she sold 19 ribeye steak dinners and 13 grilled salmon​ dinners, totaling ​$581.21. How much did each type of dinner​ cost?
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Part 1
The cost of ribeye steak dinners is ​$

enter your response here and the cost of salmon dinners is ​$

enter your response here.
​(Simplify your answer. Round to the nearest hundredth as​ needed.)

We can solve this problem using a system of linear equations. Let:

- $ x $ represent the price of a ribeye steak dinner.
- $ y $ represent the price of a grilled salmon dinner.

We are given two pieces of information:

1. On the first day, 14 ribeye steak dinners and 26 grilled salmon dinners totaled $568.69.
   \[
   14x + 26y = 568.69
   \]

2. On the second day, 19 ribeye steak dinners and 13 grilled salmon dinners totaled $581.21.
   \[
   19x + 13y = 581.21
   \]

We now have the system of equations:

\[
14x + 26y = 568.69
\]
\[
19x + 13y = 581.21
\]

To solve this, we can use either substitution or elimination. Let's use the elimination method.

### Step 1: Eliminate one variable
We will first eliminate $ y $. To do this, we need to make the coefficients of $ y $ in both equations the same. We can do this by multiplying the first equation by 13 and the second equation by 26:

\[
(14x + 26y) \cdot 13 = 568.69 \cdot 13 \implies 182x + 338y = 7392.97
\]
\[
(19x + 13y) \cdot 26 = 581.21 \cdot 26 \implies 494x + 338y = 15111.46
\]

Now we subtract the first equation from the second:

\[
(494x + 338y) - (182x + 338y) = 15111.46 - 7392.97
\]
\[
(494x - 182x) = 7718.49
\]
\[
312x = 7718.49
\]

### Step 2: Solve for $ x $
\[
x = \frac{7718.49}{312} \approx 24.74
\]

So, the cost of a ribeye steak dinner is approximately **$24.74**.

### Step 3: Solve for $ y $
Now substitute $ x = 24.74 $ back into one of the original equations. We will use the first equation:

\[
14(24.74) + 26y = 568.69
\]
\[
346.36 + 26y = 568.69
\]
\[
26y = 568.69 - 346.36
\]
\[
26y = 222.33
\]
\[
y = \frac{222.33}{26} \approx 8.55
\]

So, the cost of a grilled salmon dinner is approximately **$8.55**.

### Final Answer:
- The cost of a ribeye steak dinner is **$24.74**.
- The cost of a grilled salmon dinner is **$8.55**.

Let me know if you want more details or have any questions!

Here are five related questions to deepen your understanding:

1. What if the prices of the two dinners were swapped? How would the system of equations change?
2. How would you approach this problem if the waitress had sold an additional type of dinner?
3. Could you solve this problem graphically by plotting the two equations? How would the intersection represent the solution?
4. How would you set up the system if you had percentages rather than total dollar amounts?
5. What would happen if the numbers of dinners sold on both days were the same?

**Tip:** When solving systems of equations, elimination is often easier when you can quickly match one variable's coefficients.

A waitress sold 14 ribeye steak dinners and 26 grilled salmon dinners, totaling $568.69 on a particular day. Another day she sold 19 ribeye steak dinners and 13 grilled salmon dinners, totaling $581.21. How much did each type of dinner cost?

Learn how to solve a system of linear equations to find the cost of ribeye steak and grilled salmon dinners. This problem involves key algebraic concepts like solving systems of equations using the elimination method. Suitable for students in Grades 10-12.

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