Fundamentals of Matrix Algebra, 2011a

1.5 Applicaons of Linear Systems
(Note: while the principles outlined here will help one solve any type of problem, these
steps are wrien specifically for solving problems that involve only linear equaons.)
. Key Idea 4
Mathemacal Problem Solving
1. Understand the problem. What exactly is being
asked?
2. Idenfy the unknowns. What are you trying to find?
What units are involved?
3. Give names to your unknowns
.
(these are your variables).
4. Use the informaon given to write as many equaons
as you can that involve these variables.
5. Use the equaons to form an augmented matrix; use
Gaussian eliminaon to put the matrix into reduced
row echelon form.
6. Interpret the reduced row echelon form of the matrix
to idenfy the soluon.
7. Ensure the soluon makes sense in the context of the
problem.
Having idenfied some steps, let us put them into pracce with some examples.
. Example 17 .A concert hall has seang arranged in three secons. As part of a
special promoon, guests will recieve two of three prizes. Guests seated in the first
and second secons will receive Prize A, guests seated in the second and third secons
will receive Prize B, and guests seated in the first and third secons will receive Prize
C. Concert promoters told the concert hall managers of their plans, and asked how
many seats were in each secon. (The promoters want to store prizes for each secon
separately for easier distribuon.) The managers, thinking they were being helpful,
told the promoters they would need 105 A prizes, 103 B prizes, and 88 C prizes, and
have since been unavailable for further help. How many seats are in each secon?
S Before we rush in and start making equaons, we should be clear
about what is being asked. The final sentence asks: “How many seats are in each
secon?” This tells us what our unknowns should be: we should name our unknowns
for the number of seats in each secon. Let x 1 , x 2 and x 3 denote the number of seats
in the first, second and third secons, respecvely. This covers the first two steps of
our general problem solving technique.
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