Exam 1

1. Question

   Given the following information:

   	$+$		$+$		=	$909$
   	$+$		$+$		=	$516$
   	$+$		$+$		=	$921$

   Compute:

   $+$

   $+$

   =

   $\text{?}$

   
   
   1. $434$
   2. $378$
   3. $921$
   4. $678$
   5. $528$
   
   

   Solution

   The information provided can be interpreted as the price for three fruit baskets with different combinations of the three fruits. This corresponds to a system of linear equations where the price of the three fruits is the vector of unknowns $x$:

   $x_1=$

   $x_2=$

   $x_3=$

   The system of linear equations is then:
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{\left(\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 2\\ \hfill 2& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 2\end{array}\right)·\left(\begin{array}{c}\hfill x_1\\ \hfill x_2\\ \hfill x_3\end{array}\right)}& =\hfill & \left(\begin{array}{c}\hfill 909\\ \hfill 516\\ \hfill 921\end{array}\right)\hfill \end{array}}$

   
   This can be solved using any solution algorithm, e.g., elimination:
   

   ${\displaystyle x_1=41,\hspace{0.5em}{x}_2=53,\hspace{0.5em}{x}_3=434.}$

   
   Based on the three prices for the different fruits it is straightforward to compute the total price of the fourth fruit basket via:

   	$+$		$+$		$=$
   $x_1$	$+$	$x_2$	$+$	$x_3$	$=$
   $41$	$+$	$53$	$+$	$434$	$=$	$528$.