Chapter SH, Problem 3.23EP

To determine

To find: the number of each types of drink were sold by using crammer’s rule.

Answer to Problem 3.23EP

  $x=125;y=50$ .

Explanation of Solution

Given:

The concession stands sells at the start of game of hot chocolate for $\$1.75$ and soda for $\$1.25$ .

At the end of game $175$ drinks were sold at $\$243.75$ .

Calculation:

Let the number of hot chocolates sold are x at the price of $\$1.75$ and number of sodas sold are y at the price of $\$1.25$ .

Therefore, $x+y=175$

And the total amount is $243.75.

Therefore,

  $\begin{array}{l}1.75x+1.25y=243.5\\ \Rightarrow 7x+5y=975\end{array}$

Now use the Cramer’s Rule,

Therefore,

  $C=\left[\begin{array}{l}11\\ 75\end{array}\right]$

Now find the value of $\left|C\right|$ ,

  $\begin{array}{l}\left|C\right|=5-7\\ \left|C\right|=-2\end{array}$

Thus, the solution of above equations is,

  $\begin{array}{l}x=\frac{\left|\begin{array}{l}1175\\ 7975\end{array}\right|}{\left|C\right|}\\ x=\frac{975-7\left(175\right)}{\left|-2\right|}\\ x=\frac{975-1225}{\left(-2\right)}\\ x=125\end{array}$

Now solve to find the value of y ,

  $\begin{array}{l}y=\frac{\left|\begin{array}{l}1751\\ 9755\end{array}\right|}{\left|C\right|}\\ y=\frac{5\left(175\right)-975}{\left|-2\right|}\\ y=\frac{875-975}{\left(-2\right)}\\ y=50\end{array}$

Hence, the solution of the above equation is $x=125;y=50$ .