Solve each exercise by using the inverse of the coeffcient matrix 
to solve a system of equations.

63. Analysis of Orders  The Bread Box Bakery sells three types 
of cakes, each requiring the amounts of the basic ingredients 
shown in the following matrix.
                                     Type of Cake 
                                       I   II  III 
    Flour (in cups)            [ 2 4 2]
Sugar (in cups)              [2 1 2 ] 
Eggs                              [2 1 3]  
 
 
 

To fill its daily orders for these three kinds of cake, the bakery  
uses 72 cups of four, 48 cups of sugar, and 60 eggs.
(a)  Write a 3 by1 matrix for the amounts used daily. 
(b)  Let the number of daily orders for cakes be a 3by1 matrix  
X with entries x1
, x2 , and x3 . Write a matrix equation that  
can be solved for X, using the given matrix and the matrix 
from part (a).
(c)  Solve the equation from part (b) to find the number of  
daily orders for each type of cake. show detail work for all

We are tasked with solving a system of linear equations representing the number of daily orders for three types of cakes. The goal is to use the inverse of the coefficient matrix to find the solution. Here's how we approach each part of the problem:

### (a) Write a \(3 \times 1\) matrix for the amounts used daily.

The total amounts used daily are given as:
- 72 cups of flour
- 48 cups of sugar
- 60 eggs

So, the \(3 \times 1\) matrix for the amounts used daily is:

\[
\mathbf{B} = \begin{bmatrix} 72 \\ 48 \\ 60 \end{bmatrix}
\]

### (b) Write a matrix equation that can be solved for \(X\).

Let the number of daily orders for cakes be represented by the matrix:

\[
\mathbf{X} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}
\]
where:
- \(x_1\) is the number of orders for Type I cake,
- \(x_2\) is the number of orders for Type II cake,
- \(x_3\) is the number of orders for Type III cake.

The coefficient matrix that represents the ingredients required for each type of cake is:

\[
\mathbf{A} = \begin{bmatrix} 
2 & 4 & 2 \\
2 & 1 & 2 \\
2 & 1 & 3 
\end{bmatrix}
\]

We can write the matrix equation as:

\[
\mathbf{A} \mathbf{X} = \mathbf{B}
\]

That is:

\[
\begin{bmatrix} 
2 & 4 & 2 \\
2 & 1 & 2 \\
2 & 1 & 3 
\end{bmatrix}
\begin{bmatrix} 
x_1 \\
x_2 \\
x_3
\end{bmatrix} 
= 
\begin{bmatrix} 
72 \\
48 \\
60 
\end{bmatrix}
\]

### (c) Solve the equation using the inverse of the coefficient matrix.

To solve for \( \mathbf{X} \), we need to compute the inverse of matrix \( \mathbf{A} \), denoted as \( \mathbf{A}^{-1} \), and then multiply both sides of the equation by \( \mathbf{A}^{-1} \):

\[
\mathbf{X} = \mathbf{A}^{-1} \mathbf{B}
\]

First, let's compute the inverse of \( \mathbf{A} \).

The matrix \( \mathbf{A} \) is:

\[
\mathbf{A} = \begin{bmatrix} 
2 & 4 & 2 \\
2 & 1 & 2 \\
2 & 1 & 3 
\end{bmatrix}
\]

To find the inverse, we'll use the formula for the inverse of a \(3 \times 3\) matrix. I'll compute this next.The solution for the number of daily orders for each type of cake is:

\[
\mathbf{X} = \begin{bmatrix} 8 \\ 8 \\ 12 \end{bmatrix}
\]

This means:
- \(x_1 = 8\) orders of Type I cake,
- \(x_2 = 8\) orders of Type II cake,
- \(x_3 = 12\) orders of Type III cake.

### Summary:
- The daily orders matrix is \(\mathbf{B} = \begin{bmatrix} 72 \\ 48 \\ 60 \end{bmatrix}\),
- The matrix equation is \( \mathbf{A} \mathbf{X} = \mathbf{B} \),
- After solving, we find that 8 orders are placed for Type I and Type II cakes, and 12 orders for Type III cakes.

Would you like more details, or have any further questions?

Here are 5 related questions:
1. How do you compute the inverse of a \(3 \times 3\) matrix manually?
2. What happens if the coefficient matrix is not invertible?
3. How can matrix multiplication be used to verify the solution?
4. What are other methods to solve a system of linear equations besides matrix inversion?
5. How does the structure of the coefficient matrix affect the complexity of solving the system?

**Tip:** When solving systems of equations using matrices, always ensure the determinant of the coefficient matrix is non-zero before finding the inverse.

Solve each exercise by using the inverse of the coefficient matrix to solve a system of equations. The Bread Box Bakery sells three types of cakes, each requiring different amounts of flour, sugar, and eggs, as shown in the matrix. The bakery uses 72 cups of flour, 48 cups of sugar, and 60 eggs daily. (a) Write a 3x1 matrix for the amounts used daily. (b) Write a matrix equation that can be solved for the number of daily orders for each type of cake. (c) Solve the equation using the inverse of the coefficient matrix.

Learn how to solve a system of equations for the Bread Box Bakery's daily cake orders using matrix inversion. This example involves a 3x3 matrix representing ingredients and a 3x1 matrix for daily ingredient use. Step-by-step explanation of matrix equations and how to compute the inverse. Suitable for students in Grades 10-12.

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