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 determine how many ounces of jam each type of jar can hold, we can set up a system of linear equations based on the information provided.

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

### Step 2: Set Up the System of Equations
We are given two key pieces of information:
1. One large jar and five small jars can hold 26 ounces of jam: \( x + 5y = 26 \)
2. One large jar minus one small jar can hold 2 ounces of jam: \( x - y = 2 \)

### Step 3: Write the System of Equations
\[
\begin{aligned}
1. & \quad x + 5y = 26 \quad &\text{(Equation 1)} \\
2. & \quad x - y = 2 \quad &\text{(Equation 2)}
\end{aligned}
\]

### Step 4: Set Up the Matrices
We can express this system of equations in matrix form as follows:

#### Coefficient Matrix (A):
\[
A = \begin{pmatrix}
1 & 5 \\
1 & -1
\end{pmatrix}
\]

This matrix contains the coefficients of \( x \) and \( y \) from the system of equations.

#### Variable Matrix (X):
\[
X = \begin{pmatrix}
x \\
y
\end{pmatrix}
\]

This matrix contains the variables \( x \) and \( y \) that we want to solve for.

#### Solution Matrix (B):
\[
B = \begin{pmatrix}
26 \\
2
\end{pmatrix}
\]

This matrix contains the constants from the right-hand side of the system of equations.

### Step 5: Matrix Equation
The system of equations can now be written as a matrix equation:
\[
AX = B
\]
Which translates to:
\[
\begin{pmatrix}
1 & 5 \\
1 & -1
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
26 \\
2
\end{pmatrix}
\]

### Explanation
- The coefficient matrix \( A \) represents the coefficients of the variables \( x \) (large jar) and \( y \) (small jar) in the equations.
- The variable matrix \( X \) contains the unknowns, which are the number of ounces of jam in the large and small jars.
- The solution matrix \( B \) represents the total ounces of jam in the combination of jars as described by the problem.

This matrix equation can be solved using methods such as substitution, elimination, or matrix operations (e.g., finding the inverse of \( A \)) to determine the values of \( x \) and \( y \).

Would you like the detailed steps for solving this matrix equation, or do you have any other questions?

### Related Questions
1. How do you solve the system of equations using the substitution method?
2. How do you solve the system of equations using the elimination method?
3. What is the determinant of the coefficient matrix \( A \), and how does it affect the solution?
4. How do you find the inverse of matrix \( A \), and how is it used to solve a system of equations?
5. Can you solve the system using Cramer's Rule?
6. What if the relationship between the jars was different, e.g., \( x + 4y = 24 \)?
7. How would the solution change if the total ounces were 30 instead of 26?
8. What if the number of small jars was doubled in the first equation? How would that affect the system?

### Tip
When converting a word problem into a system of linear equations, carefully identify each condition and express it mathematically. This ensures that the system accurately represents the problem's scenario.

Learn how to solve a system of linear equations to determine the number of ounces of jam in each type of jar. This problem involves setting up coefficient matrices and solving using methods like substitution or matrix operations. Suitable for high school students studying algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation