Chapter 3, Problem 30SGR

To determine

To find: The cost of each piece of Hot Dogs, Popcorn and Soda.

Answer to Problem 30SGR

The costs of each piece of Hot Dogs, Popcorn and soda are $\text{\$}3.25,\text{\$}2.25,\text{\$}2.5$ respectively

Explanation of Solution

Given information :

Consider the environment table:

Name	Hot Dogs	Popcorn	Soda	Price
Dustin	1	2	3	$15.25
Luis	2	0	3	$14.00
Marci	1	2	1	$10.25

Calculation :

Consider the costs of each piece of Hot Dogs, Popcorn and soda are $\text{\$}x,\text{\$}y,\text{\$}z$ respectively therefore:

  $\begin{array}{l}x+2y+3z=15.25\dots \dots (1)\\ 2x+3z=\mathrm{14.......}(2)\\ x+2y+z=10.25\dots \dots (3)\end{array}$

From (2):

  $x=\frac{1}{2}(14-3z)\dots \dots (4)$

Substitute (4) in (1):

  $\begin{array}{l}\frac{1}{2}(14-3z)+2y+3z=15.25\\ 14-3z+4y+6z=30.50\\ 4y+3z=\mathrm{16.50..........}(5)\end{array}$

Substitute (4) in (3):

  $\begin{array}{c}\frac{1}{2}(14-3z)+2y+z=10.25\\ 14-3z+4y+2z=20.50\\ 4y-z=6.50\\ z=4y-6.50\dots \dots \text{(6)}\end{array}$

Substitute (6) in (5):

  $\begin{array}{l}4y+3(4y-6.50)=16.50\\ 4y+12y-19.50=16.50\\ 16y=19.50+16.50\\ 16y=36\\ y=2.25\end{array}$

Substitute $y=2.25$ in (6):

  $\begin{array}{cc}z& =4\times 2.25-6.50\\ & =9-6.5\\ & =2.5\end{array}$

Substitute $z=2.5$ in (4):

  $\begin{array}{l}x=\frac{1}{2}(14-3\times 2.5)\\ =\frac{1}{2}(14-7.5)\\ =\frac{1}{2}\times 6.5\\ =3.25\end{array}$

Therefore the costs of each piece of Hot Dogs, Popcorn and soda are $\text{\$}3.25,\text{\$}2.25,\text{\$}2.5$ respectively.