9/27/2004, LINEAR EQUATIONSMath 21b, O. KnillHOMEWORK due 9/29/2004: 1.1: 10,12,20,24,38b,26*,36* (Problems with * are recommended only).SYSTEM OF LINEAR EQUATIONS. A collection of linear equations is called asystem of linear equations.An example is3x-y-z= 0-x+ 2y-z= 0-x-y+ 3z= 9.It consists of three equations for three unknownsx, y, z.Linearmeans that no nonlinear terms likex2, x3, xy, yz3,sin(x) etc. appear. A formal definition of linearity will be given later.LINEAR EQUATION. The equationax+by=cis the general linear equation in two variables andax+by+cz=dis the general linear equation in three variables. The generallinear equationinnvariables has the forma1x1+a2x2+...+anxn= 0. Finitely many such equations form asystem of linear equations.SOLVING BY ELIMINATION.Eliminate variables.In the first example, the first equation givesz= 3x-y. Putting this into the secondand third equation gives-x+ 2y-(3x-y)= 0-x-y+ 3(3x-y)= 9or-4x+ 3y= 08x-4y= 9.The first equation givesy= 4/3xand plugging this into the other equation gives 8x-16/3x= 9 or 8x= 27which meansx= 27/8. The other valuesy= 9/2, z= 45/8 can now be obtained.SOLVE BY SUITABLE SUBTRACTION.
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