How to Solve the Puzzle: Determining the Cost of Coffee and Muffins

Let's dive into a puzzling scenario involving Maria and John at a local coffee shop. This story will introduce you to the power of algebra and the system of equations to unravel the mystery of the cost of coffee and muffins. Following the steps below will help you solve this intriguing problem and understand the underlying mathematical principles.

Understanding the Scenario

Maria and John visit a coffee shop, where they make two separate purchases. Maria buys 3 coffees and 2 muffins for a total of $13, while John buys 2 coffees and 4 muffins for $14. Our goal is to determine the cost of one coffee and one muffin. This problem can be solved using a system of equations, a crucial concept in algebra.

Solving the System of Equations

Step 1: Setting Up Equations

We start by denoting the cost of one coffee as C and the cost of one muffin as M. The purchases made by Maria and John can be described by the following equations:

Maria: 3C 2M 13 John: 2C 4M 14

Step 2: Simplification and Elimination

To solve this system, it's helpful to manipulate the equations to eliminate one variable. Let's begin by doubling the first equation:

Doubling 3C 2M 13 results in 6C 4M 26

Now, we can subtract the second equation 2C 4M 14 from this new equation:

(6C 4M) - (2C 4M) 26 - 14

This simplifies to:

4C 12

Solving for C gives:

C 12 / 4 3

Step 3: Solving for M

With the value of C known, we can substitute this into one of the original equations to solve for M. Let's use the equation 3C 2M 13 and substitute C 3 into it:

3(3) 2M 13

9 2M 13

2M 13 - 9 4

M 4 / 2 2

Conclusion and Verification

The solution to the puzzle is:

The cost of one coffee is $3. The cost of one muffin is $2.

To verify the solution, let's check both original conditions:

Verification

Maria's purchase:

3C 2M 3(3) 2(2) 9 4 13

John's purchase:

2C 4M 2(3) 4(2) 6 8 14

Both conditions are satisfied, hence the solution is correct.

Further Exploration

Understanding how to solve systems of equations is not only useful for such puzzles but also in many real-world applications. From finance to engineering, algebraic methods are essential in solving complex problems. Practice these types of equations to enhance your mathematical skills and broaden your problem-solving toolkit.