Chapter 3.7, Problem 48PPS

To determine

To find: The number of stars and number of supporting cast in the movie.

Answer to Problem 48PPS

The number of stars in the movie was $x=3$

The number of supporting cast is $y=4$

Explanation of Solution

Given information:

The salary of star is $\text{\$}5$ million and supporting cast is $\text{\$}1$ million. Total amount spent is $\text{\$}19$ million and total members are 7 members.

Calculation:

Consider the case of paying salary in a new movie. The salary of star is $\text{\$}5$ million and supporting cast is $\text{\$}1$ million. Total amount spent is $\text{\$}19$ million and total members are 7 members.

Let the number of stars be $x$

Let the number of supporting cast be $y$

Hence,

  $x+y=7$

Amount spent on stars is $5x$ and amount spent on supporting cast is $y$ .Hence

  $5x+y=19$

The set of simultaneous equations are:

  $\begin{array}{l}x+y=7\\ 5x+y=19\end{array}$

Solve equations using Cramer's method.

The coefficient matrix is,

  $\left[\begin{array}{cc}1& 1\\ 5& 1\end{array}\right]$

The determinant of coefficient matrix is,

  $\begin{array}{l}\left|C\right|=\left|\begin{array}{cc}1& 1\\ 5& 1\end{array}\right|\\ =(1\times 1-5\times 1)\\ =-4\end{array}$

Using Cramer's rule, the solution is:

  $x=\frac{\left|\begin{array}{cc}m& b\\ n& g\end{array}\right|}{\left|C\right|}$

  $=\frac{\left|\begin{array}{cc}7& 1\\ 19& 1\end{array}\right|}{-4}$

  $\begin{array}{l}=\frac{(7\times 1-19\times 1)}{-4}\\ =\frac{-12}{-4}\\ =3\end{array}$

And,

  $y=\frac{\left|\begin{array}{cc}a& m\\ f& n\end{array}\right|}{\left|C\right|}$

  $=\frac{\left|\begin{array}{cc}1& 7\\ 5& 19\end{array}\right|}{-4}$

  $\begin{array}{l}=\frac{(19\times 1-7\times 5)}{-4}\\ =\frac{-16}{-4}\\ =4\end{array}$

Hence, the solution is,

  $x=3$

  $y=4$

Therefore, the number of stars in the movie was $x=3$ and the number of supporting cast is $y=4$