Math 365: Linear Algebra September 2, 2011
Instructor: Dohse  

Project I: Systems of Equations
Due September 9, 2011

Directions: Set up both problems as a matrix or vector equation and use the questions to guide you to the answers. Be sure to remember to answer all questions and that all answers must be feasible (i.e. no negative values or fractions for any x's ). Five points will be set aside to evaluate neatness.

  1. A Problem from Chemistry
    In a chemical reaction the atoms realign themselves to form a different set of compounds without changing the number of atoms. The problem is to determine how many molecules are needed to ensure that the compounds with which you start are completely consumed by the reaction.
    When setting up the system of equations remember that the variables correspond to the number of molecules needed in the reaction, and that each element produces one equation. Also remember that your answer should be in terms of natural numbers, {1, 2, 3, ...} and that the subscripts (Ex. H2O) indicate the number of atoms (2 hydrogen atoms) in the molecule. Now balance the following reaction:
     
    Fe S + O2 → Fe2 O3 + S O2
    2. Questions:
    1. Define your four variables. (example x1 number of Fe S molecules.)
    2. State the linear equations (there should be three).
    3. What is the corresponding augmented matrix for this system of equations?
    4. What is the reduced echelon form of your augmented matrix for this system of equations?
    5. What the general solution to this problem?
    6. Determine the minimal (non-zero) solution that balances this equation.

  1. Analysis of Traffic Patterns
    By the entrance of the UNCA campus is a traffic circle which connects three roads.  The numbers in the image below give the number of cars (per hour) that enter and leave the intersection. Set up the linear system of equations, and determine the values for the traffic density for the different arcs of the circle.

    2. Traffic Circle

    Questions:
    1. How many cars are leaving the circle toward the left of this diagram?
    2. What are the six linear equations (one for each "node") of this problem?
    3. What is the corresponding augmented matrix for this system of equations?
    4. What is the reduced echelon form of your augmented matrix for this system of equations?
    5. Does this system of equations have a unique solution? Give reason for your answer?
    6. Which part of the traffic circle, (x1, x2, …), has the highest traffic density?
    7. What is the smallest possible value for x3 and still have feasible solutions?
    8. If x6= 80 what is the values of all the other x's?