One large jar and two small jars together can hold 8 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 this problem, we need to determine how many ounces of jam each large jar and small jar can hold, given two conditions.

### Step 1: Define Variables
Let:
- \( x \) = the number of ounces a large jar can hold
- \( y \) = the number of ounces a small jar can hold

### Step 2: Form the System of Equations
From the problem, we know:
1. One large jar and two small jars together can hold 8 ounces of jam:
   \[
   x + 2y = 8
   \]
2. One large jar minus one small jar can hold 2 ounces of jam:
   \[
   x - y = 2
   \]

Now, we have the following system of equations:
1. \( x + 2y = 8 \)
2. \( x - y = 2 \)

### Step 3: Matrix Representation

#### Coefficient Matrix
The coefficient matrix is made up of the coefficients of \( x \) and \( y \) in the system of equations. From the equations, the coefficient matrix is:
\[
\begin{pmatrix}
1 & 2 \\
1 & -1
\end{pmatrix}
\]

#### Variable Matrix
The variable matrix contains the unknowns \( x \) and \( y \):
\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]

#### Solution Matrix
The solution matrix contains the constants on the right-hand side of the equations:
\[
\begin{pmatrix}
8 \\
2
\end{pmatrix}
\]

### Step 4: Complete Matrix Equation
The matrix equation combining the coefficient, variable, and solution matrices is:
\[
\begin{pmatrix}
1 & 2 \\
1 & -1
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
8 \\
2
\end{pmatrix}
\]

This is the matrix setup to solve for \( x \) (the capacity of the large jar) and \( y \) (the capacity of the small jar).

Would you like me to solve this system of equations for the values of \( x \) and \( y \)?

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Here are 5 related questions:
1. How can this system be solved using the elimination method?
2. What happens if we increase the number of small jars in the first equation?
3. Can this system be solved using the substitution method?
4. What is the geometric interpretation of this system of equations?
5. How can we use matrix inversion to solve for \( x \) and \( y \)?

**Tip:** For systems of linear equations, matrix methods like Gaussian elimination or finding the inverse are powerful tools when dealing with multiple variables.

Learn how to solve a system of linear equations involving large and small jars. Understand the coefficients, variables, and solution matrices used to determine the capacity of each jar. Suitable for students in Grades 7-9.

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