082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

Solution Firstconsider the equations with a steppattern imposed as follows:
2x+3y=1 (1)
x +y =3 (2)
8.10 Gaussian elimination 287
Ouraimwillbetoperformvariousoperationsontheseequationstoremoveoreliminate
all the values underneath the step. You will probably remember from your early work
on simultaneous equations that in order to eliminate a variable from an equation, that
equationcanbemultipliedbyanysuitablenumberandthenaddedtoorsubtractedfrom
another equation. In this example we can eliminate thexterm from below the step by
multiplying the second equation by 2 and subtracting the first equation. Since the first
equationisentirelyabovethestepweshallleaveitasitstands.Thiswholeprocesswill
be writtenas follows:
R 1
R 2
→2R 2
−R 1
2x+3y=1
0x −y =5
(8.9)
where the symbolR 1
means that Equation (1) is unaltered, andR 2
→ 2R 2
−R 1
means
that Equation (2) has been replaced by 2 × Equation (2) -- Equation (1). All this may
seem to be overcomplicating a simple problem but a moment’s study of Equation (8.9)
will reveal why this ‘stepped’ form is useful. Because the value under the step is zero
wecan read offyfrom the lastline, thatis −y = 5,so that
y=−5
Knowingywe can then move up tothe first equation and substituteforytofindx.
2x +3(−5) =1
x = 8
This laststage isknown asback substitution.
Beforeweconsideranother example,let usnotesome important points:
(1) It is necessary to write down the operations used as indicated previously. This aids
checkingand provides a record ofthe working used.
(2) Theoperationsallowedtoeliminate unwanted variables are:
(a) any equation can bemultipliedby any non-zeroconstant;
(b) any equation can beadded toorsubtractedfromany other equation;
(c) equationscan beinterchanged.
It is often convenient to use matrices to carry out this method, in which case the operationsallowedarereferredtoasrowoperations.Theadvantageofusingmatricesisthat
it is unnecessary to write downx,y(and laterz) each time. To do this, wefirst form the
augmentedmatrix:
( ) 231
113
( )
( 2 3
1
so called because the coefficient matrix is augmented by the r.h.s. matrix .
1 1
3)
Itistobeunderstoodthatthisnotationmeans2x+3y = 1,andsoon,sothatwenolonger
write down x and y. Each row of the augmented matrix corresponds to one equation.