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Unformatted text preview: 8.2 Inconsistent and Dependent Systems and Their Applications 1. Apply Gaussian elimination to systems without unique solutions. Inconsistent Systems: Planes inlcrsccl nm at .a time. Three plant—‘1 are parallel With I'wn plant's arc parallel with 'I'licrc“ is no iniurwelmn paint
rm cmmmm intersection point. 1m enmmnn lﬂlct‘wi‘cllill‘a lmim eramrnnn In all three plane;
Figure 5.1 “mm plum“. may hm c rm Cmtllmm ['Iuim “I iulctwé‘licm Dependent Systems: / The planes: intersect The planes Quincirlc,
allu'mg LI eumrmm [InJ. Figure 6.2 "than; lumm may intersect ul iufimwly mung puinh
Linear systems can have one solution, no solution, or infinitely
many solutions. We can use Gaussian elimination on systems
with three or more variables to determine how many solutions
such systems have. In the case of systems with no solution or infinitely
many solutions, it is impossible to rewrite the augmented matrix in
the desired form with is down the main diagonal from upper left
to lower right, and Os below the ls.
The system has no solution if one of the rows in the rowechelon form
is a false statement, ie. 0X+0y+Oz=5.
The system has inﬁnitely many solutions if the three planes intersect
in more than one point. This is when all but one variable can be written in terms of the one variable. In a nonsquare system, the number of variables differs from
the number of equations, ie. two equations and three variables. Notice how the solution is expressed in terms of one variable. Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. 4x—2y—Z=—3
1. 7x—3y—2z=—l2
—6x+2y+2z=10 0 9 l r! 0
O 0 i L 0
0 0 —l “'2. 0 ...
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 Fall '11
 BlisinHestiyas

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