https://file.aiquickdraw.com/mathbot/file/1726234989697w7u3ef7u.jpg

The problem asks to write a system of three equations using variables $ x $, $ y $, and $ z $, representing the pounds of chicken, rice, and shellfish, respectively. We're given the following information:

- The total weight of the paella is 20 pounds.
- The total cost of the ingredients is \$32.10.
- The total amount of protein is 1120 grams.
- The table gives the following data for the ingredients:
  - Chicken: \$1.40 per pound, 110 grams of protein per pound
  - Rice: \$0.40 per pound, 20 grams of protein per pound
  - Shellfish: \$6.10 per pound, 50 grams of protein per pound

We can set up three equations based on:

1. **The total weight of the paella**:
   \[
   x + y + z = 20
   \]
   This equation represents the fact that the sum of the weights of chicken, rice, and shellfish is 20 pounds.

2. **The total cost of the paella**:
   \[
   1.40x + 0.40y + 6.10z = 32.10
   \]
   This equation represents the total cost of the chicken, rice, and shellfish.

3. **The total amount of protein**:
   \[
   110x + 20y + 50z = 1120
   \]
   This equation represents the total amount of protein from the chicken, rice, and shellfish.

Thus, the system of equations is:

\[
\begin{aligned}
1. & \quad x + y + z = 20 \\
2. & \quad 1.40x + 0.40y + 6.10z = 32.10 \\
3. & \quad 110x + 20y + 50z = 1120
\end{aligned}
\]

Would you like help solving this system of equations?

Here are 5 related questions to expand the concept:
1. How can you solve a system of equations with three variables using substitution?
2. What methods can be used to solve this system of equations, such as elimination or matrix methods?
3. How do changes in the price of ingredients affect the total cost equation?
4. What happens if the total weight of the paella changes? How would the equations be adjusted?
5. How can you interpret the results of solving this system of equations in a real-world catering scenario?

**Tip**: In any system of equations, organizing your known values and understanding the relationships between variables is key to constructing accurate models.

A caterer combines ingredients to make a paella, a Spanish fiesta dish. The paella weighs 20 lb, costs $32.10, and supplies 1120 g of protein. Let x represent the number of pounds of chicken, y represent the number of pounds of rice, and z represent the number of pounds of shellfish used by the caterer. Complete parts (a) and (b) below.
(a) Write a system of three equations to find the weight of each ingredient that the caterer uses. Write the first equation about the total weight of the paella.

Learn how to solve a system of three equations to determine the weight of each ingredient in a paella dish. The problem involves algebraic concepts such as linear equations and systems of equations. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation