Question

1. Your school's Key Club decided to sell fruit baskets to raise

money for a local charity. The club sold a total of 80 fruit
baskets. There were three different types of fruit baskets.
Small fruit baskets sold for $15.75 each, medium fruit baskets
sold for $25 each, and large fruit baskets sold for $32.50
each. The Key Club took in a total of 52086.25. and they sold
twice as many large baskets as small baskets.
a. Formulate a system of three linear equations in three
variables to represent this problem situation. Be sure to
define your variables.
b. Solve the system of three linear equations using
technology with matrices. Write your answer in terms of
the problem situation.
c. Describe the solution in terms of the problem situation.
2. Is it always easier to solve a system of linear equations using
matrices? Explain your reasoning.

Answer 1

The system of equations for the fruit basket sales problem is solved using matrices, resulting in 15 small fruit baskets, 20 medium fruit baskets, and 45 large fruit baskets sold.

Step-by-step explanation:

a. Let x represent the number of small fruit baskets sold, y represent the number of medium fruit baskets sold, and z represent the number of large fruit baskets sold. The system of equations can be formulated as follows:

x + y + z = 80

15.75x + 25y + 32.50z = 52086.25

z = 2x

b. Solving the system of equations using matrices, we get:

x = 15, y = 20, z = 45

c. The solution means that 15 small fruit baskets, 20 medium fruit baskets, and 45 large fruit baskets were sold.

Answer 2

Let the small , medium and large baskets be denoted by s,m & l

Formulating the equations

s+m+l= 80--------- equation1

$15.75s+ $25 m + $32.5l = $2086.25------equation2

2s=l---------equation 3

Using the technology s=15, m=35 and l= 30

15+35+30= 80--------- equation1

$15.75(15)+ $25 (35)+ $32.5 (30) = $2086.25------equation2

2(15)=(30)---------equation 3

Yes because multiple variables can be easily calculated through matrices using technology

In Matrix form

`\left[\begin{array}{ccc}1&1&1\\15.75&25&32.5\\-2&0&1\end{array}\right]`
`\left[\begin{array}{c}s\\m\\l\end{array}\right]` =
`\left[\begin{array}{c} 80\\2086.25\\0\end{array}\right]`

Let A=
`\left[\begin{array}{ccc}1&1&1\\15.75&25&32.5\\-2&0&1\end{array}\right]`

Z=
`\left[\begin{array}{c}s\\m\\l\end{array}\right]`

B =
`\left[\begin{array}{c} 80\\2086.25\\0\end{array}\right]`

Z= A⁻¹ B

A⁻¹= A mod/ det A

det A= -1

Using the technology s=15, m=35 and l= 30

15 small baskets,

35 medium baskets

30 large baskets were sold.

Part C:

15+35+30= 80--------- equation1

$15.75(15)+ $25 (35)+ $32.5 (30) = $2086.25------equation2

236.25+ 875+ 975=$2086.25

$2086.25=$2086.25

2(15)=(30)---------equation 3

Answer 2:

Yes because multiple variables can be easily calculated using technology.

N.B There is a typing error in question its $2086.25 not 52086.25