## Exam 1

1. #### Question

Given the following information: $+$ $+$ = $564$ $+$ $+$ = $873$ $+$ $+$ = $864$

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} 2 & 0 & 1 \\ 1 & 0 & 2 \\ 0 & 1 & 2 \end{array} \right) \cdot \left( \begin{array}{r} x_1 \\ x_2 \\ x_3 \end{array} \right) & = & \left( \begin{array}{r} 564 \\ 873 \\ 864 \end{array} \right) \end{aligned} This can be solved using any solution algorithm, e.g., elimination: $x_1 = 85, \, x_2 = 76, \, x_3 = 394.$ Based on the three prices for the different fruits it is straightforward to compute the total price of the fourth fruit basket via: $+$ $+$ = $x_1$ $+$ $x_2$ $+$ $x_3$ = $85$ $+$ $76$ $+$ $394$ = $555$