a. First, let's define our variables, as stated:
Let \( x \) be the cost per pound of pistachios, and let \( y \) be the cost per pound of cashews.
Now, based on the information given, we can write two equations to represent the costs that Keith and Tracey paid.
Keith's purchase gives us the first equation:
Keith paid $39 for 3 pounds of pistachios and 2 pounds of cashews. This can be written as:
\[ 3x + 2y = 39 \]
Tracey's purchase gives us the second equation:
Tracey paid $23 for 2 pounds of pistachios and 1 pound of cashews. This can be written as:
\[ 2x + y = 23 \]
We now have the system of equations:
\[ 3x + 2y = 39 \]
\[ 2x + y = 23 \]
b. To write the matrix equation, we can represent the system of equations as a linear algebra equation of the form \( A \cdot X = B \), where:
- \( A \) is the coefficient matrix,
- \( X \) is the variable matrix,
- \( B \) is the constants matrix.
For our system of equations, the matrix representation is:
\[ A = \begin{bmatrix} 3 & 2 \\ 2 & 1 \end{bmatrix}, X = \begin{bmatrix} x \\ y \end{bmatrix}, B = \begin{bmatrix} 39 \\ 23 \end{bmatrix} \]
Then, our matrix equation is:
\[ \begin{bmatrix} 3 & 2 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 39 \\ 23 \end{bmatrix} \]
To solve for \( X \), we find the inverse of matrix \( A \) and multiply it by matrix \( B \). Without going into the details of how to calculate the inverse of a 2x2 matrix or the mechanics of matrix multiplication, we'll assume that the matrix operations are done correctly to yield the solution for \( X \), which represents the values of \( x \) and \( y \).
After solving these operations, we find that:
\[ x = 7 \]
\[ y = 9 \]
This means the cost of a pound of pistachios (x) is $7.00 and the cost of a pound of cashews (y) is $9.00. These are the rates at which Keith and Tracey would be contributing to their respective costs for pistachios and cashews.
