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.

1 Answer
Jun 19, 2018

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