Two student clubs were selling t-shirts and school notebooks to raise money for an upcoming school event. In the first few minutes, club A sold 2 t-shirts and 3 notebooks, and made $40. Club B sold 1 t-shirt and 1 notebook, for a total of $16.
A matrix with 2 rows and 2 columns, where row 1 is 2 and 3 and row 2 is 1 and 1, is multiplied by matrix with 2 rows and 1 column, where row 1 is x and row 2 is y, equals a matrix with 2 rows and 1 column, where row 1 is 40 and row 2 is 16.
Use matrices to solve the equation and determine the cost of a t-shirt and the cost of a notebook. Show or explain all necessary steps.
Let's start by setting up the equations using matrices.
Let's assume the cost of a t-shirt is x and the cost of a notebook is y.
For Club A:
2t-shirts + 3notebooks = $40
For Club B:
1 t-shirt + 1 notebook = $16
We can write this as a matrix equation:
A * X = B
Where A is a 2x2 matrix:
| 2 3 |
| 1 1 |
X is a 2x1 matrix:
| x |
| y |
And B is a 2x1 matrix:
| 40 |
| 16 |
We need to find the values of x and y.
To solve the equation, we need to find the inverse of matrix A and multiply it on both sides of the equation.
A^-1 * A * X = A^-1 * B
Since A^-1 * A gives us the identity matrix, the equation simplifies to:
X = A^-1 * B
Let's calculate the inverse of matrix A:
A^-1 = (1 / (ad - bc)) * | d -b |
| -c a |
a = 2, b = 3, c = 1, d = 1
So, a^-1 = (1 / (2*1 - 3*1)) * | 1 -3 |
| -1 2 |
a^-1 = (1 / -1) * | 1 -3 |
| -1 2 |
a^-1 = | -1 3 |
| 1 -2 |
Now, let's calculate A^-1 * B:
| -1 3 | | 40 | | (-1*40) + (3*16) | | -40 + 48 | | 8 |
| 1 -2 | * | 16 | = | (1*40) + (-2*16) | = | 40 - 32 | = | 8 |
So, X = | 8 |
| 8 |
Therefore, the cost of a t-shirt is $8 and the cost of a notebook is $8.
We have the equation:
Matrix A * Matrix B = Matrix C
Where Matrix A is a 2x2 matrix representing the quantities of t-shirts and notebooks sold by each club, Matrix B is a 2x1 matrix representing the cost of each item, and Matrix C is a 2x1 matrix representing the total revenue from each club.
From the given information, we can set up the matrices as follows:
Matrix A:
| 2 3 |
| 1 1 |
Matrix B:
| x |
| y |
Matrix C:
| 40 |
| 16 |
To solve for x and y, we need to find the inverse of Matrix A and multiply it by Matrix C.
Inverse of Matrix A, denoted by A^(-1), can be found as follows:
A^(-1) = (1/(ad - bc)) * adj(A)
Where a, b, c, and d are the elements of Matrix A and adj(A) is the adjugate of Matrix A.
Calculating the inverse of Matrix A:
ad - bc = (2 * 1) - (3 * 1) = 2 - 3 = -1
A^(-1) = (1/(-1)) * adj(A)
adj(A) is obtained by swapping the elements (2, 3) and (1, 1) and changing their signs:
adj(A) = | 1 -3 |
| -1 2 |
So, A^(-1) = -1 * | 1 -3 |
| -1 2 |
= | -1 3 |
| 1 -2 |
Now, we can multiply A^(-1) by Matrix C to find the values of x and y:
Matrix A^(-1) * Matrix C = | -1 3 | * | 40 |
| 1 -2 | | 16 |
Multiplying the matrices:
|-1 * 40 + 3 * 16 |
| 1 * 40 - 2 * 16 |
Simplifying:
| -40 + 48 |
| 40 - 32 |
= | 8 |
| 8 |
So, the matrix | 8 | represents the values of x and y, where x is the cost of a t-shirt and y is the cost of a notebook.
Therefore, the cost of a t-shirt is $8, and the cost of a notebook is also $8.
To solve this problem using matrices, let's define the matrices involved:
Matrix A:
[2 3]
Matrix B:
[x]
[y]
Matrix C:
[40]
[16]
Now, we have the equation:
Matrix A * Matrix B = Matrix C
Multiplying these matrices can be done by multiplying the corresponding elements and summing them up:
2x + 3y = 40 (Equation 1)
x + y = 16 (Equation 2)
Now we have a system of two equations with two variables. We can solve this system to find the values of x and y.
One way to solve this system is by the method of substitution. Let's solve Equation 2 for either x or y and substitute it into Equation 1. We'll solve Equation 2 for x:
x = 16 - y
Now substitute x into Equation 1:
2(16 - y) + 3y = 40
Expand and simplify:
32 - 2y + 3y = 40
Combine like terms:
y = 8
Now substitute y = 8 back into Equation 2:
x + 8 = 16
Subtract 8 from both sides:
x = 8
So, the value of x is 8, and the value of y is 8.
Now we have the values of x and y, which represent the prices of a t-shirt and a notebook, respectively. The cost of a t-shirt is $8, and the cost of a notebook is also $8.