Mathematics-Online course: Prepcourse Mathematics-Linear Algebra and Geometry-Systems of Linear Equations-Elimination of Variables in Linear Systems of Equations

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	Mathematics-Online course: Prepcourse Mathematics - Linear Algebra and Geometry - Systems of Linear Equations
	Elimination of Variables in Linear Systems of Equations

A linear equation

$\displaystyle a_1x_1+a_2x_2+\dots=f$

with $a_1\neq 0$ can be tranformed into

$\displaystyle x_1=(f-a_2x_2-\cdots)/a_1\,.$

By inserting such expressions into the remaining equations, linear systems can thus successively be reduced into systems in fewer variables.

(Authors: Höllig/Abele)

Consider the following linear system of equations:

$\begin{displaymath} \begin{array}{rcrcl} 2x & - & 3y &=& 1 \\ 5x & - & 4y &=& 6 \end{array}\end{displaymath}$

The first line yields

$\displaystyle x=\frac{1+3y}{2}$

and by insertion into the second equation it then follows

$\displaystyle 5\cdot\frac{1+3y}{2}-4y=6 \quad\Leftrightarrow\quad \frac{7}{2}\,y+\frac{5}{2}=6 \quad\Leftrightarrow\quad y=1\,.$

Therewith it finally is

$\displaystyle x=\frac{1+3\cdot 1}{2}=2\,.$

(Authors: Höllig/Abele)

automatically generated 9/18/2007