Chapter 6.5, Problem 5CYU

a.

To determine

To write the system of linear equations based on the information of the graph.

Answer to Problem 5CYU

  $-x+y=-11$ and $12.6+21y=978.6$

Explanation of Solution

Formula Used:

Equation of a line passing through two point $(x_1,y_1)$ and $(x_2,y_2)$ is:

  $\frac{x-x_1}{x_1-x_2}=\frac{y-y_1}{y_1-y_2}$

Calculation:

Blue Line: This line passes through the points (11, 0) and (20, 9).

Hence, the equation of the line will be:

  $\begin{array}{l}\Rightarrow \frac{x-11}{11-20}=\frac{y-0}{0-9}\\ \Rightarrow \frac{x-11}{-9}=\frac{y-0}{-9}\\ \Rightarrow y=x-11\\ \Rightarrow -x+y=-11\end{array}$

Red Line: This line passes through (0, 46.6) and (21, 34).

Hence, the equation of the line will be:

  $\begin{array}{l}\Rightarrow \frac{x-0}{0-21}=\frac{y-46.6}{46.6-34}\\ \Rightarrow \frac{x-0}{-21}=\frac{y-46.6}{12.6}\\ \Rightarrow 12.6x=-21y+978.6\\ \Rightarrow 12.6x+21y=978.6\end{array}$

Conclusion:

Hence, the system of equations is: $-x+y=-11$ and $12.6+21y=\mathrm{978.6.}$

b.

To determine

To interpret the meaning of each equation

Explanation of Solution

Given:

  $-x+y=-11$

  $12.6+21y=978.6$

Let,

  $-x+y=-\mathrm{11..............}(i)$

  $12.6+21y=\mathrm{978.6...................}(ii)$

Equation ( i ), explains the increase of increase of total profit with increase of the number of subs.

Equation ( ii ), explains the decrease of increase of total number of items that can be sold with increase of the number of subs.

c.

To determine

To solve the system of equations and explain the meaning of the solution.

Answer to Problem 5CYU

  $x=36$and $y=25.$

Explanation of Solution

Given:

  $-x+y=-11$

  $12.6+21y=978.6$

Calculation:

Let,

  $-x+y=-\mathrm{11..............}(i)$

  $12.6+21y=\mathrm{978.6...................}(ii)$

Now, add equation ( ii ) with $12.6\times$ equation ( i ),

  $\begin{array}{l}\Rightarrow 12.6(-x+y)+(12.6x+21y)=12.6\times (-11)+978.6\\ \Rightarrow -12.6x+12.6y+12.6x+21y=-138.6+978.6\\ \Rightarrow 33.6y=840\\ \Rightarrow y=25\end{array}$

Putting the value of y in equation ( i ),

  $\begin{array}{l}\Rightarrow -x+25=-11\\ \Rightarrow x=36\end{array}$

Hence, for blue line, when there are 36 subs, there will be total profit of 25.

For red line, when there are 36 subs, the total items that could be sold are 25.

Conclusion:

Hence the solution is: $x=36$and $y=25.$