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Linear Systems of Equations in Two Variables

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A linear system of equations in two unknowns ("variables") $ x$ and $ y$ has the form

\begin{displaymath} \begin{array}{rcrcl} ax & + & by &=& f \\ cx & + & dy &=& g\,. \end{array}\end{displaymath}

The system has a unique solution for all $ (f,g)$ , iff

$\displaystyle \Delta = ad-bc\neq 0\,. $

The solution

$\displaystyle x=\frac{df-bg}{\Delta}\,,\qquad\quad y=\frac{ag-cf}{\Delta} $

can be interpreted as the intersection of the lines given by the two equations.
\includegraphics[width=6cm]{lgsgeraden}
If $ \Delta=0$, the lines are parallel to one another. In this case either no solution exists or - if the lines coincide - there are infinitely many solutions.
(Authors: Höllig/Abele)
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  automatically generated 9/18/2007