Exam 1

1. Question

   Given the following information:

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

   Compute:

   $+$

   $+$

   =

   $\text{?}$

   
   

   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: $$\begin{aligned} \left( \begin{array}{rrr} 1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \end{array} \right) \cdot \left( \begin{array}{r} x_1 \\ x_2 \\ x_3 \end{array} \right) & = & \left( \begin{array}{r} 909 \\ 516 \\ 921 \end{array} \right) \end{aligned}$$ This can be solved using any solution algorithm, e.g., elimination: $$x_1 = 41, \, x_2 = 53, \, x_3 = 434.$$ Based on the three prices for the different fruits it is straightforward to compute the total price of the fourth fruit basket via:

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