Chapter 6.3, Problem 13PPE

a)

To determine

To find: The system of equations to model the given situation is,

Answer to Problem 13PPE

Required system of equations to model given situation is,

  $\begin{array}{l}12x+2y=90\mathrm{......}(i)\\ 6x+2y=60\mathrm{.....}(ii)\end{array}$

Explanation of Solution

Given information: Show will last 90 min and will include 12 solo and 2 ensemble acts. Another show lasting 60 min will include 6 solo and 2 ensemble acts.

Concept used:

Let each solo is performed for x minutes and ensemble act for y minutes. And form two linear equations in these two variables based on two given conditions. Then solve them together using elimination method. An equation is formed using following statement of equality,

  $\text{Number of solo}\times \text{ its minutes + number of ensemple}\times \text{ its minutes = total minutes}$

Calculation: Based on first condition given, first equation is formed as,

  $12x+2y=90\mathrm{.....}(i)$

Using second condition, second equation is formed as,

  $6x+2y=60\mathrm{.....}(ii)$

Conclusion: Above two equations form the system of linear equations to model the given situation.

b)

To determine

To find: The duration of each solo act as well as ensemble act.

Answer to Problem 13PPE

Duration of each solo act is 5 minutes and of each ensemble act is 15 minutes.

Explanation of Solution

Given information: Two linear equations modeling given situations as obtained in part (a) are,

  $\begin{array}{l}12x+2y=90\mathrm{......}(i)\\ 6x+2y=60\mathrm{.....}(ii)\end{array}$

Concept used: As the difference of coefficients of y terms in both equations is zero in given equations, so solve them using elimination method to get the values of x and y.

Calculation:

Subtract second equations from first equation and solve for x as,

  $\begin{array}{l}12x+2y-6x-2y=90-60\\ \Rightarrow 6x=30\\ \Rightarrow x=\frac{30}{6}=5\end{array}$

Use second equation,

  $\begin{array}{l}6\times 5+2y=60\\ 30+2y=60\\ 2y=60-30=30\\ \Rightarrow y=\frac{30}{2}=15\end{array}$

Conclusion: So, solo act will be played for 5 minutes each and ensemble act will be played for 15 minutes each.