rainbow Harbor Cruises charges $16/adult and $8/child for a round trip ticket. The records show that on a certain weekend, 1200 people took the cruise on saturady and 800 people took the cruise on sunday, The total receipte for saturday were $13,600 and the total receipts for sunday were $9,600. Determine how many adults and children took the cruise on saturday and sunday set up two systems o linear equations and use the inverse of the coefficient matrices to answwer the question

Using the inverse of a Matrix:

Let ‘a’ be adults and ‘c’ be children
a + b = 1000
16 a + 8b = 12,800

1 1 1000 -16 R1 (R2) 1 1 1000 R2 / 8
16 8 12800 0 -8 -3200

1 1 1000 -1 R2 + R1 1 0 600
0 1 400 0 1 400

Therefore, there were 600 adults and 400 children on Saturday.

SUNDAY

a + b = 800
16a + 8 b = 9600

1 1 1000 -16 R1 (R2) 1 1 1000 R2 / 8
16 8 12800 0 -8 -3200

1 1 1000 -1 R2 + R1 1 0 400
0 1 400 0 1 400

Therefore there were 400 adults and 400 children on Saturday.

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Let's assume that the number of adults on Saturday is represented by A1, the number of children on Saturday is represented by C1, the number of adults on Sunday is represented by A2, and the number of children on Sunday is represented by C2.

From the given information, we can set up the following two linear equations:

Equation 1: 16A1 + 8C1 = 13,600 (Total receipts for Saturday)

Equation 2: 16A2 + 8C2 = 9,600 (Total receipts for Sunday)

To solve these equations using the inverse of the coefficient matrices, we can rewrite them in matrix form:

Equation 1: [16 8] * [A1] = [13,600]
[C1]

Equation 2: [16 8] * [A2] = [9,600]
[C2]

To find the inverse of the coefficient matrix [16 8], we need to calculate its determinant first:

Determinant = (16 * 8) - (8 * 16) = 0

Since the determinant is zero, the coefficient matrix [16 8] does not have an inverse. Therefore, we cannot use the inverse of the coefficient matrices to solve this system of linear equations.

However, we can still solve these equations using other methods, such as substitution or elimination.

To solve this problem using two systems of linear equations, we need to assign variables to represent the number of adults and children on each day.

Let's say 'a' represents the number of adults and 'c' represents the number of children on Saturday, and 'x' represents the number of adults and 'y' represents the number of children on Sunday.

From the given information, we can set up the following equations:

For Saturday:
16a + 8c = 13,600 (equation 1)

For Sunday:
16x + 8y = 9,600 (equation 2)

Now we have a system of two linear equations. We can solve this system using the inverse of the coefficient matrices.

First, we need to write these equations in matrix form. The coefficient matrix is:

A = [[16, 8], [16, 8]]

The variable matrix is:

X = [[a], [c]] for Saturday
X = [[x], [y]] for Sunday

And the constant matrix is:

B = [[13,600], [9,600]]

Now we can find the inverse of matrix A.

The inverse of A, denoted as A^(-1), can be calculated as:

A^(-1) = 1 / (ad - bc) * [[d, -b], [-c, a]]

where:
a = 16, b = 8
c = 16, d = 8

Calculating the determinant (ad - bc), we have:

ad - bc = (16 * 8) - (8 * 16)
= 128 - 128
= 0

Since the determinant is 0, the inverse of matrix A does not exist.

Therefore, we cannot use the inverse of the coefficient matrices to solve this system of equations. We need to use another method, such as substitution or elimination, to find the solution.