Solve Equations 9x-2y+z=50,x+5y-3z=18,-2x+2y+7z=19 using Relaxation method

Solution:
Total Equations are `3`

`9x-2y+z=50`

`x+5y-3z=18`

`-2x+2y+7z=19`


The residuals from equations, we get
`R_1=50-9x+2y-z`

`R_2=18-x-5y+3z`

`R_3=19+2x-2y-7z`

The table for operation is

	`R_1`	`R_2`	`R_3`
`deltax`	-9	-1	2
`deltay`	2	-5	-2
`deltaz`	-1	3	-7

Solution steps are
`1^(st)` Approximation

`R_1=50-0=50`

`R_2=18-0=18`

`R_3=19-0=19`

Maximum is `R_1=50`

`deltax=50/9=5.5556`

`2^(nd)` Approximation

`R_1=50-9*5.5556=50-50=0`

`R_2=18-1*5.5556=18-5.5556=12.4444`

`R_3=19-(-2)*5.5556=19--11.1111=30.1111`

Maximum is `R_3=30.1111`

`deltaz=30.1111/7=4.3016`

`3^(rd)` Approximation

`R_1=0-1*4.3016=0-4.3016=-4.3016`

`R_2=12.4444-(-3)*4.3016=12.4444--12.9048=25.3492`

`R_3=30.1111-7*4.3016=30.1111-30.1111=0`

Maximum is `R_2=25.3492`

`deltay=25.3492/5=5.0698`

`4^(th)` Approximation

`R_1=-4.3016-(-2)*5.0698=-4.3016--10.1397=5.8381`

`R_2=25.3492-5*5.0698=25.3492-25.3492=0`

`R_3=0-2*5.0698=0-10.1397=-10.1397`

Maximum is `R_3=-10.1397`

`deltaz=-10.1397/7=-1.4485`

`5^(th)` Approximation

`R_1=5.8381-1*-1.4485=5.8381--1.4485=7.2866`

`R_2=0-(-3)*-1.4485=0-4.3456=-4.3456`

`R_3=-10.1397-7*-1.4485=-10.1397--10.1397=0`

Maximum is `R_1=7.2866`

`deltax=7.2866/9=0.8096`

`6^(th)` Approximation

`R_1=7.2866-9*0.8096=7.2866-7.2866=0`

`R_2=-4.3456-1*0.8096=-4.3456-0.8096=-5.1552`

`R_3=0-(-2)*0.8096=0--1.6192=1.6192`

Maximum is `R_2=-5.1552`

`deltay=-5.1552/5=-1.031`

`7^(th)` Approximation

`R_1=0-(-2)*-1.031=0-2.0621=-2.0621`

`R_2=-5.1552-5*-1.031=-5.1552--5.1552=0`

`R_3=1.6192-2*-1.031=1.6192--2.0621=3.6813`

Maximum is `R_3=3.6813`

`deltaz=3.6813/7=0.5259`

`8^(th)` Approximation

`R_1=-2.0621-1*0.5259=-2.0621-0.5259=-2.588`

`R_2=0-(-3)*0.5259=0--1.5777=1.5777`

`R_3=3.6813-7*0.5259=3.6813-3.6813=0`

Maximum is `R_1=-2.588`

`deltax=-2.588/9=-0.2876`

`9^(th)` Approximation

`R_1=-2.588-9*-0.2876=-2.588--2.588=0`

`R_2=1.5777-1*-0.2876=1.5777--0.2876=1.8653`

`R_3=0-(-2)*-0.2876=0-0.5751=-0.5751`

Maximum is `R_2=1.8653`

`deltay=1.8653/5=0.3731`

`10^(th)` Approximation

`R_1=0-(-2)*0.3731=0--0.7461=0.7461`

`R_2=1.8653-5*0.3731=1.8653-1.8653=0`

`R_3=-0.5751-2*0.3731=-0.5751-0.7461=-1.3212`

Maximum is `R_3=-1.3212`

`deltaz=-1.3212/7=-0.1887`

`11^(th)` Approximation

`R_1=0.7461-1*-0.1887=0.7461--0.1887=0.9349`

`R_2=0-(-3)*-0.1887=0-0.5662=-0.5662`

`R_3=-1.3212-7*-0.1887=-1.3212--1.3212=0`

Maximum is `R_1=0.9349`

`deltax=0.9349/9=0.1039`

`12^(th)` Approximation

`R_1=0.9349-9*0.1039=0.9349-0.9349=0`

`R_2=-0.5662-1*0.1039=-0.5662-0.1039=-0.6701`

`R_3=0-(-2)*0.1039=0--0.2077=0.2077`

Maximum is `R_2=-0.6701`

`deltay=-0.6701/5=-0.134`

`13^(th)` Approximation

`R_1=0-(-2)*-0.134=0-0.268=-0.268`

`R_2=-0.6701-5*-0.134=-0.6701--0.6701=0`

`R_3=0.2077-2*-0.134=0.2077--0.268=0.4758`

Maximum is `R_3=0.4758`

`deltaz=0.4758/7=0.068`

`14^(th)` Approximation

`R_1=-0.268-1*0.068=-0.268-0.068=-0.336`

`R_2=0-(-3)*0.068=0--0.2039=0.2039`

`R_3=0.4758-7*0.068=0.4758-0.4758=0`

Maximum is `R_1=-0.336`

`deltax=-0.336/9=-0.0373`

`15^(th)` Approximation

`R_1=-0.336-9*-0.0373=-0.336--0.336=0`

`R_2=0.2039-1*-0.0373=0.2039--0.0373=0.2412`

`R_3=0-(-2)*-0.0373=0-0.0747=-0.0747`

Maximum is `R_2=0.2412`

`deltay=0.2412/5=0.0482`

`16^(th)` Approximation

`R_1=0-(-2)*0.0482=0--0.0965=0.0965`

`R_2=0.2412-5*0.0482=0.2412-0.2412=0`

`R_3=-0.0747-2*0.0482=-0.0747-0.0965=-0.1712`

Maximum is `R_3=-0.1712`

`deltaz=-0.1712/7=-0.0245`

`17^(th)` Approximation

`R_1=0.0965-1*-0.0245=0.0965--0.0245=0.1209`

`R_2=0-(-3)*-0.0245=0-0.0734=-0.0734`

`R_3=-0.1712-7*-0.0245=-0.1712--0.1712=0`

Maximum is `R_1=0.1209`

`deltax=0.1209/9=0.0134`

`18^(th)` Approximation

`R_1=0.1209-9*0.0134=0.1209-0.1209=0`

`R_2=-0.0734-1*0.0134=-0.0734-0.0134=-0.0868`

`R_3=0-(-2)*0.0134=0--0.0269=0.0269`

Maximum is `R_2=-0.0868`

`deltay=-0.0868/5=-0.0174`

`19^(th)` Approximation

`R_1=0-(-2)*-0.0174=0-0.0347=-0.0347`

`R_2=-0.0868-5*-0.0174=-0.0868--0.0868=0`

`R_3=0.0269-2*-0.0174=0.0269--0.0347=0.0616`

Maximum is `R_3=0.0616`

`deltaz=0.0616/7=0.0088`

`20^(th)` Approximation

`R_1=-0.0347-1*0.0088=-0.0347-0.0088=-0.0435`

`R_2=0-(-3)*0.0088=0--0.0264=0.0264`

`R_3=0.0616-7*0.0088=0.0616-0.0616=0`

Maximum is `R_1=-0.0435`

`deltax=-0.0435/9=-0.0048`


Solution By Relaxation Method.
`x=sum deltax=6.1528~=6.15`

`y=sum deltay=4.3087~=4.31`

`z=sum deltaz=3.2425~=3.24`

Iterations are tabulated as below

Iteration	Operation	`deltax` (9)	`deltay` (5)	`deltaz` (7)	`R_1`	`R_2`	`R_3`
1	`x=y=z=0`	0	0	0	50	18	19
2	`deltax=50/9=5.5556`	5.5556	0	0	0	12.4444	30.1111
3	`deltaz=30.1111/7=4.3016`	0	0	4.3016	-4.3016	25.3492	0
4	`deltay=25.3492/5=5.0698`	0	5.0698	0	5.8381	0	-10.1397
5	`deltaz=-10.1397/7=-1.4485`	0	0	-1.4485	7.2866	-4.3456	0
6	`deltax=7.2866/9=0.8096`	0.8096	0	0	0	-5.1552	1.6192
7	`deltay=-5.1552/5=-1.031`	0	-1.031	0	-2.0621	0	3.6813
8	`deltaz=3.6813/7=0.5259`	0	0	0.5259	-2.588	1.5777	0
9	`deltax=-2.588/9=-0.2876`	-0.2876	0	0	0	1.8653	-0.5751
10	`deltay=1.8653/5=0.3731`	0	0.3731	0	0.7461	0	-1.3212
11	`deltaz=-1.3212/7=-0.1887`	0	0	-0.1887	0.9349	-0.5662	0
12	`deltax=0.9349/9=0.1039`	0.1039	0	0	0	-0.6701	0.2077
13	`deltay=-0.6701/5=-0.134`	0	-0.134	0	-0.268	0	0.4758
14	`deltaz=0.4758/7=0.068`	0	0	0.068	-0.336	0.2039	0
15	`deltax=-0.336/9=-0.0373`	-0.0373	0	0	0	0.2412	-0.0747
16	`deltay=0.2412/5=0.0482`	0	0.0482	0	0.0965	0	-0.1712
17	`deltaz=-0.1712/7=-0.0245`	0	0	-0.0245	0.1209	-0.0734	0
18	`deltax=0.1209/9=0.0134`	0.0134	0	0	0	-0.0868	0.0269
19	`deltay=-0.0868/5=-0.0174`	0	-0.0174	0	-0.0347	0	0.0616
20	`deltaz=0.0616/7=0.0088`	0	0	0.0088	-0.0435	0.0264	0
	Total	6.1528	4.3087	3.2425

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