System of Equations: An Introduction
Many books on linear algebra will introduce matrices via systems of linear
equations. We tried a different approach. We hope this way you will appreciate
matrices as a powerful tool useful not only to solve linear systems of
equations. Basically, the problem of finding some unknowns linked to each others
via equations is called a system of equations.
For example,
and
are systems of two equations with two unknowns (x and y), while
is a system of two equations with three unknowns (x, y, and z).
These systems of equations occur naturally in many real life problems. For
example, consider a nutritious drink which consists of whole egg, milk, and
orange juice. The food energy and protein for each of the ingredients are given
by the table:
A natural question to ask is how much of each ingredient do we need to produce a
drink of 540 calories and 25 grams of protein. In order to answer that, let x
be the number of eggs, y the amount of milk (in cups), and z the
amount of orange of juice (in cups). Then we need to have
The task of Solving a system consists of finding the unknowns, here:
x, y and z. A solution is a set of numbers once substituted
for the unknowns will satisfy the equations of the system. For example, (2,1,2)
and
(0.325, 2.25, 1.4) are solutions to the system above.
The fundamental problem associated to any system is to find all the
solutions. One way is to study the structure of its set of solutions which, in
some cases, may help finding the solutions. Indeed, for example, in order to
find the solutions to a linear system, it is enough to find just a few of them.
This is possible because of the rich structure of the set of solutions.
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