Question

Write a system of equations to represent this problem and determine the unit price of each item purchased? Define your variables.

Alvin, Theodore, and Simon went to the movies. Alvin bought 2 boxes of popcorn, 4 cherry slushies, and 2 boxes of candy. He spent $49.50. Theodore bought 3 boxes of popcorn, 2 cherry slushies, and 4 boxes of candy. He spent $57.75. Simon bought 3 boxes of popcorn, 3 cherry slushies, and 1 box of candy. He spent $38.50.

Answer 1

The cost of each box of popcorn is `$ 3.75`;
The cost of each cherry sushi is `$6.25`; and
The cost of each box of candy is `$ 8.5`.

Explanation:

Alvin, Theodore, and Simon went to the movies. Alvin bought 2 boxes of popcorn, 4 cherry sushies, and 2 boxes of candy. He spent $49.50. Theodore bought 3 boxes of popcorn, 2 cherry sushies, and 4 boxes of candy. He spent $57.75. Simon bought 3 boxes of popcorn, 3 cherry sushies, and 1 box of candy. He spent $38.50.

Let the cost of each box of popcorn be `x`;
Let the cost of each cherry sushi be `y`; and
Let the cost of each box of candy be `z`.

Given That :
Alvin bought 2 boxes of popcorn, 4 cherry sushies, and 2 boxes of candy. He spent $49.50.

`therefore 2x + 4y + 2z = $ 49.50` -------------equation (1)

Theodore bought 3 boxes of popcorn, 2 cherry sushies, and 4 boxes of candy. He spent $57.75.

`therefore 3x +2y + 4z = $ 57.75` ---------------equation(2)

Simon bought 3 boxes of popcorn, 3 cherry sushies, and 1 box of candy. He spent $38.50.

`therefore 3x +3y + 1z = $ 38.50`-------------- equation(3)

The set of equations with three variables to solve is:
`2x + 4y + 2z = $ 49.50` ------------- (1)
`3x +2y + 4z = $ 57.75` --------------(2)
`3x +3y + 1z = $ 38.50`--------------(3)

We can solve this set of three equations by elimination and substitution method.

Consider equations (2) and (3) to eliminate `x`:

Subtract (3) from (2). That gives:

(2) - (3) `=> 0x - 1y + 3z = $ 19.25`

`=> -y +3z = 19.25`------------equation (4)

Consider equation (1) and (3) to eliminate `x`:
(1) x 3 - (3) x 2 will give:

`=> 0x + 6y +4z = 148.5 - 77 = 71.5`

`=> 6y +4z = 71.5` ------------(5)

Now consider (4) and (5) to eliminate `y`,

(4) x 6 + (5) gives:

`22z = 115.5 +71.5 = 187`

`=> z= 8.5`

`therefore z= 8.5`

Substitute value of `z` in (5) to find `y`:

`=> 6y +4xx 8.5 = 71.5`

`=> y = (71.5 - 34)/ 6`

`y = 6.25`

`therefore y = 6.25`

Substitute value of `y` and `z` in equation (1):

`(1)=> 2x + 4y + 2z = $ 49.50`

`=> 2x +4 xx 6.25 +2 xx 8.5 = 49.50`

`=> 2x = 49.50 - 25 - 17`

`=> 2x = 7.5`

`=> x = 3.75`
`therefore x= $3.75, y = $6.25 and z= $8.5`
Cross check by substituting in (2)
`=> 3x +2y + 4z = $ 57.75`
`=> 3 (3.75 ) + 2( 6.25) + 4( 8.5) = 11.25 + 12.5 + 34 = 57.7`