# 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 $\implies - y + 3 z = 19.25$------------equation (4) Consider equation (1) and (3) to eliminate $x$: (1) x 3 - (3) x 2 will give: $\implies 0 x + 6 y + 4 z = 148.5 - 77 = 71.5$$\implies 6 y + 4 z = 71.5$------------(5) Now consider (4) and (5) to eliminate $y$, (4) x 6 + (5) gives: $22 z = 115.5 + 71.5 = 187$$\implies z = 8.5$$\therefore z = 8.5$Substitute value of $z$in (5) to find $y$: $\implies 6 y + 4 \times 8.5 = 71.5$$\implies y = \frac{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

$\implies 2 x + 4 \times 6.25 + 2 \times 8.5 = 49.50$

$\implies 2 x = 49.50 - 25 - 17$

$\implies 2 x = 7.5$

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