Question

A shelter receives a shipment of items worth $1040. Bags of dog food are valued at $5 each, flea collars at $6 each, and dog toys at $2 each. There are 4 times as many bags of food as collars. The number of collars and toys together equals 100. How many of each item are in the shipment? What is the system of equations? What is the augmented matrix?​

Answer 1

The shelter's shipment problem can be represented by a system of linear equations, which can be solved using an augmented matrix. The equations consider the relationship between bags of dog food, flea collars, and dog toys, constrained by their total value and quantity.

Step-by-step explanation:

To solve the shelter's shipment problem, we will define the variables and set up a system of equations. Let x be the number of flea collars, 4x the number of bags of dog food, and y the number of dog toys. We are given the following information:

• The value of all items is $1040.
• There are 4 times as many bags of food as collars.
• The number of collars and toys together equals 100.

From this, we get the equations:

1. 5(4x) + 6x + 2y = 1040 (value equation)
2. 4x = 4x (bags of food to collars equation)
3. x + y = 100 (collars and toys equation)

By simplifying the value equation, we get:

20x + 6x + 2y = 1040

26x + 2y = 1040

Then the system of equations is:

1. 26x + 2y = 1040
2. x + y = 100

To form the augmented matrix, we rewrite the system in matrix form:

[ [26, 2 | 1040] [1, 1 | 100]]