Chapter 6.7, Problem 50HP

To determine

To write and solve: word problem represented by the given matrix

Answer to Problem 50HP

Here, the system is $x=3.5$ and the Download charge for each latest song is $\$2.5$ and for each album is $\$3.5$ .

Explanation of Solution

Given: $\left[\begin{array}{l}3\text{ 1 13}\\ \text{2 1 9}\text{.5}\end{array}\right]$

Calculation:

Objective: To write and solve word problem represented by $\left[\begin{array}{l}3\text{ 1 13}\\ \text{2 1 9}\text{.5}\end{array}\right]$ .

Problem: An online music class offers individual songs for one- price or entire albums for another. A pays $\$13$ to download one individual latest song and three albums. D pays $\$9.5$ to download one individual latest song and two albums.

(a) How much does the music latest song be $\$y$ and download charge for each album be $\$x$ .

(b) How much does the music club charge to download an entire album?

Let download charge for each latest song be $\$y$ and download charge for each album be $\$x$ .

Hence,

A pays: $\left(3\times x\right)+\left(1\times y\right)$

D pays: $\left(2\times x\right)+\left(1\times y\right)$

Therefore from the given data

  $\begin{array}{l}\left(3\times x\right)+\left(1\times y\right)=13\Rightarrow \text{3}x+y=13\\ \left(2\times x\right)+\left(1\times y\right)=9.5\Rightarrow \text{2}x+y=9.5\end{array}$

Augmented matrix for the given system is

  $\left[\begin{array}{l}3\text{ 1 | 13}\\ \text{2 1 | 9}\text{.5}\end{array}\right]$

Solve the above system

  $\begin{array}{l}3x+y=13\Rightarrow \text{ 3}x\text{ + y = 13}\\ \text{2}x+y=9.5\text{ Multiply by}\left(-1\right)\Rightarrow -\text{2}x-\text{ y =}-\text{9}\text{.5}\\ \text{ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\\ \text{ Add }x=3.5\\ \Rightarrow x=3.5\end{array}$

Put the value of $x$ in first equation

  $\begin{array}{l}\Rightarrow 3\left(3.5\right)+y=13\\ \Rightarrow 10.5+y=13\\ \Rightarrow y=13-10.5=2.5\end{array}$

Hence

(a) Download charge for each latest song is $\$2.5$ .

(b) Download charge for each album is $\$3.5$ .

Conclusion:

Therefore, the system is $x=3.5$ and Download charge for each latest song is $\$2.5$ and for each album is $\$3.5$ .