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 | | | |
This material is intended as a summary. Use your textbook for detail explanation.
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