Chapter 3, Problem 30SGR

To determine

To calculate: The cost of Hot Dogs, Popcorn, Soda purchased by Dustin, Luis and Marci.

Answer to Problem 30SGR

The cost of Hot Dogs, Popcorn, Soda are $\$3.25,\$2.25\text{ and }\$2.50$ respectively.

Explanation of Solution

Given information:

The number of snacks and price paid by Dustin, Luis and Marci is provided below,

  $\begin{array}{ccccc}\text{Name}& \text{Hot Dogs}& \text{Popcorn}& \text{Soda}& \text{Price}\\ \text{Dustin}& 1& 2& 3& \$15.25\\ \text{Luis}& 2& 0& 3& \$14.00\\ \text{Marci}& 1& 2& 1& \$10.25\end{array}$

Calculation:

It is provided that the number of snacks and price paid by Dustin, Luis and Marci is provided below,

  $\begin{array}{ccccc}\text{Name}& \text{Hot Dogs}& \text{Popcorn}& \text{Soda}& \text{Price}\\ \text{Dustin}& 1& 2& 3& \$15.25\\ \text{Luis}& 2& 0& 3& \$14.00\\ \text{Marci}& 1& 2& 1& \$10.25\end{array}$

Let price of hot dogs be x , popcorn be y and soda be z .

Consider the system of equations

  $\begin{array}{c}x+2y+3z=15.25\\ 2x+3z=14\\ x+2y+z=10.25\end{array}$

Take the first and the third equation,

  $\begin{array}{c}x+2y+3z=15.25\\ x+2y+z=10.25\end{array}$

Use the method of elimination to solve the system above.

Subtract both the equations,

  $\begin{array}{ccccccc}x& +& 2y& +& 3z& =& 15.25\\ x& +& 2y& +& z& =& 10.25\\ -& & & & & & \\ & & & +& 2z& =& 5\end{array}$

Therefore,

  $\begin{array}{c}2z=5\\ z=2.5\end{array}$

Now substitute $z=2.5$ in the equation $2x+3z=14$ ,

  $\begin{array}{c}2x+3\left(2.5\right)=14\\ x=3.25\end{array}$

Now substitute $z=2.5$ and $x=3.25$ in the equation $x+2y+3z=15.25$ ,

  $\begin{array}{c}3.25+2y+3\left(2.5\right)=15.25\\ b=2.25\end{array}$

Therefore, solution is the coordinate pair, $\left(3.25,2.25,2.50\right)$.

Thus, the cost of Hot Dogs, Popcorn, Soda are $\$3.25,\$2.25\text{ and }\$2.50$ respectively.