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,

$\begin{displaymath}\left\{ \begin{array}{lll} 2x - 3y &=& 1\\ x+3y &=& -2\\ \end{array} \right.\end{displaymath}$

and

$\begin{displaymath}\left\{ \begin{array}{lll} x^2 +y^2 &=& 1\\ x+3y &=& -2\\ \end{array} \right.\end{displaymath}$

are systems of two equations with two unknowns (x and y), while

$\begin{displaymath}\left\{ \begin{array}{lll} 2x - 3y^2 &=& -1\\ x+y + z &=& 1\\ \end{array} \right.\end{displaymath}$

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:

$\begin{displaymath}\begin{array}{l\vert c\vert c} &\mbox{Food Energy} & \mbox{Pr... ...0 & 9 \\ 1\; \mbox{cup orange juice} & 110 & 2 \\ \end{array}\end{displaymath}$

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

$\begin{displaymath}\left\{ \begin{array}{lll} 80x + 160y + 110z &=& 540\\ 6x+9y + 2z &=& 25\\ \end{array} \right.\end{displaymath}$

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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