One large jar and five small jars can hold 26 ounces of jam. One large jar minus one small jar can hold 2 ounces of jam. 

Set up the coefficient matrix, variable matrix, and solution matrix that could be used to determine how many ounces of jam are in each type of jar. Show or explain all necessary steps.




To solve the problem of determining how many ounces of jam are in each type of jar, we can set up a system of linear equations based on the given information.

### Step 1: Define Variables
- Let \( x \) represent the number of ounces of jam in a large jar.
- Let \( y \) represent the number of ounces of jam in a small jar.

### Step 2: Set Up the Equations
Given the information:
1. One large jar and five small jars together hold 26 ounces of jam.
   \[
   x + 5y = 26
   \]
2. One large jar minus one small jar holds 2 ounces of jam.
   \[
   x - y = 2
   \]

### Step 3: Set Up the Coefficient Matrix, Variable Matrix, and Solution Matrix
These equations can be represented in matrix form as:

1. **Coefficient Matrix**:
   \[
   \mathbf{A} = \begin{pmatrix} 
   1 & 5 \\ 
   1 & -1 
   \end{pmatrix}
   \]

2. **Variable Matrix**:
   \[
   \mathbf{X} = \begin{pmatrix} 
   x \\ 
   y 
   \end{pmatrix}
   \]

3. **Solution Matrix**:
   \[
   \mathbf{B} = \begin{pmatrix} 
   26 \\ 
   2 
   \end{pmatrix}
   \]

### Step 4: Matrix Equation
The system of equations can be written as a single matrix equation:

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

Which is:

\[
\begin{pmatrix} 
1 & 5 \\ 
1 & -1 
\end{pmatrix}
\begin{pmatrix} 
x \\ 
y 
\end{pmatrix}
=
\begin{pmatrix} 
26 \\ 
2 
\end{pmatrix}
\]

This matrix equation represents the system of linear equations that can be solved to find the values of \( x \) and \( y \), which represent the number of ounces of jam in the large and small jars, respectively.

Would you like to proceed with solving this system of equations, or do you have any other questions?

### Related Questions

1. How can the system of linear equations be solved using the substitution method?
2. What would the system of equations look like if the large jar held 4 more ounces than five small jars together?
3. How can the determinant of the coefficient matrix \(\mathbf{A}\) help determine the uniqueness of the solution?
4. What happens if the system of equations has no solution? How can this be detected?
5. How would you interpret the solution graphically?
6. Can the matrix equation be solved using Gaussian elimination? What would the steps look like?
7. If one large jar and two small jars held 14 ounces of jam, how would this affect the system of equations?
8. How can the inverse of the coefficient matrix be used to solve the system?

#### Tip:
When setting up matrix equations from word problems, carefully define each variable and ensure the equations accurately represent the given conditions. This clarity is essential for correctly solving the system.

Explore how to solve a system of linear equations involving large and small jars of jam. Understand the setup of coefficient matrix, variable matrix, and solution matrix to determine the ounces of jam in each type of jar. Suitable for Grades 9-12.

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