Question

URGENT! HELP PLS :)

Question 3 (Essay Worth 4 points)

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 $20. Club B sold 2 t-shirts and 1 notebook, for a total of $8.

A matrix with 2 rows and 2 columns, where row 1 is 2 and 3 and row 2 is 2 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 20 and row 2 is 8.

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.

Answer 1

The given matrix equation can be written as:

[2 3; 2 1] * [x; y] = [20; 8]

Multiplying the matrices on the left side of the equation gives us the system of equations:

2x + 3y = 20 2x + y = 8

To solve for x and y using matrices, we can use the inverse matrix method. First, we need to find the inverse of the coefficient matrix [2 3; 2 1]. The inverse of a 2x2 matrix [a b; c d] can be calculated using the formula: (1/(ad-bc)) * [d -b; -c a].

Let’s apply this formula to our coefficient matrix:

The determinant of [2 3; 2 1] is (21) - (32) = -4. Since the determinant is not equal to zero, the inverse of the matrix exists and can be calculated as:

(1/(-4)) * [1 -3; -2 2] = [-1/4 3/4; 1/2 -1/2]

Now we can use this inverse matrix to solve for x and y. Multiplying both sides of our matrix equation by the inverse matrix gives us:

[-1/4 3/4; 1/2 -1/2] * [2x + 3y; 2x + y] = [-1/4 3/4; 1/2 -1/2] * [20; 8]

Solving this equation gives us:

[x; y] = [0; 20/3]

So, a t-shirt costs $0 and a notebook costs $20/3.