Answer:
To determine how many adults and children you can invite to the skating party within the given budget, we can use an inverse matrix. Let's set up the problem as a system of equations.
Let:
x = number of adults to invite
y = number of children to invite
We can form two equations based on the given information:
Equation 1: Cost of admission for adults: 3.50x
Equation 2: Cost of admission for children: 2.25y
We also have the constraint that the total number of people (adults and children) should not exceed 20:
x + y ≤ 20
To solve this system of equations, we can represent it in matrix form:
[3.50 2.25] [x] [50]
[y]
Let's call the coefficient matrix A, the variable matrix X, and the constant matrix B:
A = [3.50 2.25]
X = [x]
[y]
B = [50]
To find the solution, we can use the inverse matrix of A:
A^-1 = [a b]
[c d]
where a, b, c, and d are the elements of the inverse matrix.
The solution is given by X = A^-1 * B:
X = [a b] [50]
[c d]
Multiplying A^-1 and B, we get:
[a b] [50] [solution for x]
[c d] = [solution for y]
Once we determine the values for x and y, we will know how many adults and children you can invite within the given budget.
Please note that I have used approximate values for the admission costs.