We are given a system of equations related to the bakery's daily orders for three types of cakes. The bakery uses specific amounts of flour, sugar, and eggs for each type of cake, and we need to determine the number of daily orders for each cake using matrix methods.

Part (a): Write a $3 \times 1$ Matrix for the Amounts Used Daily

The bakery uses:

• 72 cups of flour,
• 48 cups of sugar,
• 60 eggs.

We can represent this as a $3 \times 1$ matrix:

$\mathbf{B} = \begin{pmatrix} 72 \\ 48 \\ 60 \end{pmatrix}$

Part (b): Write a Matrix Equation

Let the number of daily orders for the three types of cakes (Cake I, Cake II, and Cake III) be represented by the matrix:

$\mathbf{X} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$

Where:

• $x_1$ is the number of orders for Cake I,
• $x_2$ is the number of orders for Cake II,
• $x_3$ is the number of orders for Cake III.

The given matrix of ingredients required for each cake is:

$A = \begin{pmatrix} 2 & 2 & 2 \\ 4 & 1 & 1 \\ 2 & 2 & 3 \end{pmatrix}$

Where each entry corresponds to the amount of each ingredient (flour, sugar, eggs) required for each type of cake.

The matrix equation that can be solved for $\mathbf{X}$ is:

$A \mathbf{X} = \mathbf{B}$

Substituting the known values:

$\begin{pmatrix} 2 & 2 & 2 \\ 4 & 1 & 1 \\ 2 & 2 & 3 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 72 \\ 48 \\ 60 \end{pmatrix}$

Part (c): Solve the Matrix Equation

To solve for $\mathbf{X}$, we first need to check if the matrix $A$ is invertible. If it is, we can solve for $\mathbf{X}$ using the formula:

$\mathbf{X} = A^{-1} \mathbf{B}$

Let’s calculate the inverse of matrix $A$ and solve for $\mathbf{X}$.The solution for the number of daily orders for each type of cake is:

$\mathbf{X} = \begin{pmatrix} 4 \\ 44 \\ -12 \end{pmatrix}$

This means:

• $x_1 = 4$ orders for Cake I,
• $x_2 = 44$ orders for Cake II,
• $x_3 = -12$, which doesn't make sense physically, as negative orders are not possible.

This negative value suggests that either there is an error in the data provided, or the bakery's supply constraints cannot be met with this combination of cakes. A revision of the problem might be necessary.

Let me know if you'd like to explore this further or make adjustments to the problem.

Here are 5 related questions to consider:

1. What does it mean if a system of equations yields a negative solution?
2. How do we interpret cases where the system has no realistic solutions?
3. Can we modify the constraints to achieve non-negative orders for each cake type?
4. How do we handle systems with more constraints than variables?
5. What real-world situations might lead to similar infeasibility in optimization problems?

Tip: When solving real-world problems with systems of equations