A local hardware store sells three types of boxes; small, medium and large. The cost of the boxes are; small $10, medium $20 and large $30. In the previous week a total of 1200 boxes were sold resulting in proceeds of $20 000 for the week. The number of small boxes sold was equal to the sum of the medium and large boxes sold.
Use matrix methods to determine how many of each type of box was sold.
To solve this problem using matrix methods, we can set up a system of equations. Let's define the variables as follows:
Let x represent the number of small boxes sold.
Let y represent the number of medium boxes sold.
Let z represent the number of large boxes sold.
Now, we can set up the equations based on the given information:
1) The total number of boxes sold: x + y + z = 1200
2) The total proceeds from the sale of boxes: 10x + 20y + 30z = 20000
3) The number of small boxes sold is equal to the sum of the medium and large boxes sold: x = y + z
To solve this system of equations using matrix methods, we need to represent it in matrix form.
Let's set up the coefficient matrix A, variable matrix X, and the constant matrix B:
Coefficient matrix A:
1 1 1
10 20 30
1 -1 -1
Variable matrix X:
x
y
z
Constant matrix B:
1200
20000
0
Now we can write the system of equations in matrix form as AX = B.
To solve for X, we can multiply both sides of the equation by the inverse of matrix A:
(A^(-1)) * AX = (A^(-1)) * B
Since matrix A is not a square matrix, it does not have an inverse. However, we can solve the system of equations using Gaussian elimination or other methods.
Alternatively, we can use matrix software, such as MATLAB or Python's numpy library, to solve this system of equations numerically.