The Mountain Trail Resort sold 64 condominiums last year, all 1, 2, or 3 bedroom condominiums. The total revenue from the sale of the condominiums was $5,493,000. The combined square footage of all the condominiums sold was 51,544 square feet. One bedroom condominiums sold for $72,000, two bedroom condominiums sold for $87,000, and three bedroom condominiums sold for $102,000. One bedroom condominiums have an area of 512 square feet, two bedroom condominiums have an area of 840 square feet, and three bedroom condominiums have an area of 1120 square feet. How many of each size condominium did Mountain Trail Resort sell last year? Use the matrices A and B below to enter the matrices that you use to solve this problem. Your solution must include a system of equations with a definition of the variables and an explanation of how you solved the problem using matrices.
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[ | | ] =A
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[ | ] =B
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To solve this problem using matrices, we need to define our variables and set up a system of equations.
Let's define the variables:
x = number of one-bedroom condominiums sold
y = number of two-bedroom condominiums sold
z = number of three-bedroom condominiums sold
We can now set up our system of equations based on the given information:
1. The total number of condominiums sold:
x + y + z = 64
2. The total revenue from the sale of condominiums:
72000x + 87000y + 102000z = 5493000
3. The total square footage of all the condominiums sold:
512x + 840y + 1120z = 51544
To solve this system of equations using matrices, we can represent it in matrix form as follows:
A * [x y z] = B
where A is a 3x3 matrix representing the coefficients of the variables, [x y z] is a column matrix representing the variables, and B is a column matrix representing the constants.
The matrices A and B can be constructed as follows:
A = [[1 1 1]
[72000 87000 102000]
[512 840 1120]]
B = [[64]
[5493000]
[51544]]
Now, to find the values of x, y, and z, we can solve the system of equations using matrix algebra:
A * [x y z] = B
1. Find the inverse of matrix A: A^(-1)
2. Multiply both sides by A^(-1):
A^(-1) * A * [x y z] = A^(-1) * B
3. Simplify the left side:
[x y z] = A^(-1) * B
By performing matrix multiplication, we can find the values of x, y, and z.
Finally, substitute these values back into the original equations to determine the number of each size condominium Mountain Trail Resort sold last year.