## 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:
 $\begin{array}{ccc}\multicolumn{1}{c}{\left(\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 2\\ \hfill 2& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 2\end{array}\right)·\left(\begin{array}{c}\hfill {x}_{1}\\ \hfill {x}_{2}\\ \hfill {x}_{3}\end{array}\right)}& =\hfill & \left(\begin{array}{c}\hfill 909\\ \hfill 516\\ \hfill 921\end{array}\right)\hfill \end{array}$

This can be solved using any solution algorithm, e.g., elimination:
 ${x}_{1}=41, {x}_{2}=53, {x}_{3}=434.$

Based on the three prices for the different fruits it is straightforward to compute the total price of the fourth fruit basket via: $+$ $+$ $=$ ${x}_{1}$ $+$ ${x}_{2}$ $+$ ${x}_{3}$ $=$ $41$ $+$ $53$ $+$ $434$ $=$ $528$.