Question

A store sells two different fruit baskets with mangos and kiwis. The first basket has 2 mangos and 3 kiwis for $9.00. The second basket has 5 mangos and 2 kiwis for $14.25. Find the cost of each type of fruit.

a. Explain how you would write a system of equations to represent the information given.
b. Write the system of equations as a matrix.
c. Find the identity and inverse matrices for the coefficient matrix.
d. Use the inverse to solve the system.
e. Interpret your answer in this situation.

Answer 1

By setting up a system of linear equations using the quantities and prices of the fruit baskets, converting it to matrix form, and finding the inverse of the coefficient matrix, we can solve for the individual costs of mangos and kiwis.

Step-by-step explanation:

To solve the problem of finding the cost of each type of fruit using the information provided by the store, we can set up a system of linear equations based on the quantities and total costs of the fruit baskets.

1. Let m be the cost of one mango and k be the cost of one kiwi.
2. The first basket's equation is 2m + 3k = 9 (two mangos and three kiwis for $9.00).
3. The second basket's equation is 5m + 2k = 14.25 (five mangos and two kiwis for $14.25).

We can write this system as a matrix:

[[2, 3], [5, 2]] represents the coefficients and [9, 14.25] represents the total costs.

To find the inverse matrix, we calculate the inverse of the coefficient matrix. Once we have the inverse, we can use it to solve the system by multiplying the inverse matrix by the column of total costs.

This will give us the individual costs of mangos and kiwis, which represent the values of m and k. The results tell us how much the store charges for each mango and each kiwi.