Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. Every day, a cupcake store sells 5,000 cupcakes in chocolate and vanilla flavors. If the chocolate flavor is 3 times as popular as the vanilla flavor, how many of each cupcake sell per day?

Step-by-Step Solution

Answer

3750 chocolate and 1250 vanilla cupcakes are sold per day.

Step 1: Identify Variables

Let the number of chocolate cupcakes sold per day be represented by \( x \), and the number of vanilla cupcakes be represented by \( y \).

Step 2: Set Up Equations

From the problem, we have two key pieces of information:1. The total number of chocolate and vanilla cupcakes sold per day is 5,000: \( x + y = 5000 \).2. Chocolate cupcakes are three times as popular as vanilla: \( x = 3y \).

Step 3: Form the Augmented Matrix

Using the equations \( x + y = 5000 \) and \( x = 3y \), the augmented matrix is written as: \[\begin{bmatrix}1 & 1 & | & 5000 \1 & -3 & | & 0\end{bmatrix}\]

Step 4: Use Row Reduction

To solve the matrix, perform row operations. Start by subtracting the second row from the first row:\[\begin{bmatrix}1 & 1 & | & 5000 \1 & -3 & | & 0\end{bmatrix}\rightarrow\begin{bmatrix}0 & 4 & | & 5000 \1 & -3 & | & 0\end{bmatrix}\]

Step 5: Solve the Simplified Matrix

From the matrix:1. Solve \( 4y = 5000 \) leading to \( y = 1250 \).2. Substitute \( y = 1250 \) into \( x = 3y \), giving \( x = 3 \times 1250 = 3750 \).

Step 6: Conclusion

The solution shows that the store sells 3750 chocolate cupcakes and 1250 vanilla cupcakes per day.

Key Concepts

Row Reduction

Row reduction is a method used to simplify systems of linear equations. It involves performing operations on the rows of the augmented matrix to reach a simpler form, typically the row-echelon or reduced row-echelon form. The goal is to make it easier to solve for each variable step by step.

During row reduction, you can perform three types of operations:

• Swap two rows.
• Multiply a row by a non-zero scalar.
• Add or subtract multiples of one row from another row.

The main idea is to systematically eliminate variables from specific rows, ultimately allowing us to directly read off the solutions to the system of equations. In the exercise, row reduction helps simplify the augmented matrix until it is possible to solve for the unknowns: using substitution, we find that 1250 vanilla and 3750 chocolate cupcakes are sold per day.

System of Equations

A system of equations is a set of equations with multiple variables that you solve simultaneously. In this context, the system of equations models a real-world problem and requires an approach such as substitution, elimination, or matrix methods for a solution.

For example, the cupcake problem is modeled by:

• Equations representing the total cupcakes: \( x + y = 5000 \)
• The popularity ratio between chocolate and vanilla: \( x = 3y \)

To find how many cupcakes of each type are sold per day, we solve the set of linear equations. The first equation represents the sum of both types of cupcakes, while the second provides a relationship between them. These two pieces of information, when combined, give a complete depiction of the situation, allowing us to solve for each unknown using matrix operations like row reduction.

Matrix Operations

Matrix operations are fundamental tools in linear algebra, used to solve systems of equations efficiently. In the exercise provided, matrices help organize our equations and perform systematic operations to find our solution.

The key operations involved include:

• Constructing the augmented matrix from given equations.
• Performing row operations (part of the row reduction process) to simplify the matrix.
• Reading solutions directly from the simplified matrix.

Converting a set of linear equations into matrix form is useful because it allows application of algorithms that computers can execute efficiently. The augmented matrix combines the coefficients of variables with the constants from each equation in a streamlined form, here \[\begin{bmatrix}1 & 1 & | & 5000 \1 & -3 & | & 0\end{bmatrix}\]Using matrix operations, one can solve complex systems that would be cumbersome to handle with traditional methods.