Systems of Linear Equations: End of Systems of Linear Equations
End of system of linear equations
In this chapter we have introduced a widely used technique: solving systems of linear equations.
Many models that describe practical situations, are constructed by assuming dependencies between variables. Because even large systems of linear equations can be solved very efficient, these models are often still easy to calculate.
In this chapter we have dealt with very small systems. Larger systems will be discussed in the course Linear Algebra.
We have also shown how systems of equations with two unknowns are an algebraic representation of two lines in the flat plane. A summary of the algebraic and geometrical world, categorized to three types of solutions, are shown in the following table.
| algebraic | number of solutions | geometrical | example | Example solution |
| regular | #1# | two intersecting lines | #\lineqs{x+y&=&1\cr 2x+y&=&1\cr}\phantom{longggg}# | #x=0\land y = 1# |
| dependent #\ # | #\infty# | one line (described twice) | #\lineqs{x+y&=&1\cr 2x+2y&=&2\cr}# | #x=1-y# |
| contrary | #0# | two parallel lines | #\lineqs{x+y&=&1\cr 2x+2y&=&1\cr}# | #none# |
This issue is further elaborated on in the chapter 2-dimensional geometry: points and lines.
The next step in complexity after linear equations are quadratic equations, in which both terms with #x# and with #x^2# occur. In some exercices already we encountered such equations, as part of a system. The simplest case of one quadratic equation with one unknown is discussed in the chapter Quadratic equations, which is a natural sequel to this chapter.
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