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Question Number 46592 by Tawa1 last updated on 29/Oct/18

$$\mathrm{Find}\mathrm{a}\mathrm{solution}\mathrm{to}:7\mathrm{x}+5\mathrm{y}+15\mathrm{z}+12\mathrm{w}=149$$

Commented by Tawa1 last updated on 29/Oct/18

$$\mathrm{No}\mathrm{general}\mathrm{solution}\mathrm{sir}$$

Answered by MrW3 last updated on 30/Oct/18

$$7x+5y+15z+12w=149$$$$withx,y,z,w\in \mathbb{Z}$$$$gcd(7,5,15,12)=1whichdivides149,$$$$\Rightarrow solutionexists.$$$$$$$$theoriginaleqn.canbeformedinto:$$$$7x+5(y+3z)+12w=149$$$$or$$$$y+3z=u...\left(iii\right)$$$$7x+5u=v...\left(ii\right)$$$$v+12w=149...\left(i\right)$$$$$$$$eqn.\left(i\right):$$$$v=5andw=12isaparticularsolution,$$$$\Rightarrow v=5+12m$$$$\Rightarrow w=12-m$$$$$$$$eqn.\left(ii\right):$$$$7x+5u=1hasaparticularsolution$$$$x=3,u=-4$$$$itsgeneralsolutionis$$$$x=3+5i,u=-4-7i$$$$thegeneralsolutionfor$$$$7x+5u=visthen$$$$x=(3+5i)v=(3+5i)(5+12m)=15+36m+5i(5+12m)=15+36m+5n$$$$u=(-4-7i)v=(-4-7i)(5+12m)=-20-48m-7i(5+12m)=-20-48m-7n$$$$$$$$eqn.\left(i\right):$$$$y+3z=1hasaparticularsolution$$$$y=-2,z=1$$$$itsgeneralsolutionis$$$$y=-2+3j,z=1-j$$$$thegeneralsolutionfor$$$$y+3z=uisthen$$$$y=(-2+3j)u=(-2+3j)(-20-48m-7n)=40+96m+14n+3j(-20-48m-7n)=40+96m+14n+3k$$$$z=(1-j)u=(1-j)(-20-48m-7n)=-20-48m-7n-j(-20-48m-7n)=-20-48m-7n-k$$$$$$$$\Rightarrow \{\begin{array}{l}x=15+36m+5n\\ y=40+96m+14n+3k\\ z=-20-48m-7n-k\\ w=12-m\end{array}$$$$withm,n,k\in \mathbb{Z}$$

Commented by Tawa1 last updated on 29/Oct/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by Tawa1 last updated on 30/Oct/18

$$\mathrm{Please}\mathrm{sir},\mathrm{i}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{get}:$$$$\mathrm{x}=(3+5\mathrm{i})\mathrm{how}\mathrm{you}\mathrm{got}\mathrm{it}(3+5\mathrm{i})$$$$\mathrm{u}=(-4-7\mathrm{i})\mathrm{how}\mathrm{you}\mathrm{got}\mathrm{it}(-4-7\mathrm{i})$$$$\mathrm{y}=(-2+3\mathrm{j})\mathrm{how}\mathrm{you}\mathrm{got}\mathrm{it}(-2+3\mathrm{j})$$$$\mathrm{z}=(1-\mathrm{j})\mathrm{how}\mathrm{you}\mathrm{got}\mathrm{it}(1-\mathrm{j})$$$$$$$$\mathrm{I}\mathrm{found}\mathrm{those}\mathrm{steps}\mathrm{difficult}\mathrm{to}\mathrm{understand}\mathrm{sir}.$$$$\mathrm{I}\mathrm{will}\mathrm{be}\mathrm{happy}\mathrm{if}\mathrm{you}\mathrm{can}\mathrm{break}\mathrm{it}\mathrm{down}\mathrm{sir}.$$$$\mathrm{Thanks}\mathrm{for}\mathrm{every}\mathrm{time}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by MrW3 last updated on 30/Oct/18

$$whatIcanexplainisexplainedinQ46157.$$$$besidesIhaveputmorestepsinto$$$$myworkingabove.$$$$Ordoyoumeanyoujusthavegenerally$$$$problemtofindaparticularsolution$$$$foreachoftheequations?$$

Commented by Tawa1 last updated on 30/Oct/18

$$\mathrm{Wow},\mathrm{i}\mathrm{understand}\mathrm{better}\mathrm{now}\mathrm{with}\mathrm{the}\mathrm{steps}\mathrm{you}\mathrm{add}\mathrm{in}\mathrm{your}$$$$\mathrm{solution},\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by Tawa1 last updated on 30/Oct/18

$$\mathrm{from}\mathrm{equation}\left(\mathrm{ii}\right)\mathrm{sir}.\mathrm{you}\mathrm{type}\mathrm{mistake}\mathrm{but}\mathrm{corrected}\mathrm{it}\mathrm{in}\mathrm{next}\mathrm{line}$$$$\mathrm{x}=3+5\mathrm{i},\mathrm{u}=-4-7\mathrm{i}.$$$$$$$$\mathrm{But}\mathrm{sir},\mathrm{why}\mathrm{from}\mathrm{u}=(-4-7\mathrm{i})\mathrm{v}=.......=-20-48\mathrm{m}-7\mathrm{i}(5+12\mathrm{m})$$$$$$$$\mathrm{Now},-7\mathrm{i}(5+12\mathrm{m})\mathrm{becomes}7\mathrm{n}.\mathrm{Reason}\mathrm{sir}.$$

Commented by MrW3 last updated on 30/Oct/18

$$\mathrm{i}(5+12\mathrm{m})isanintegerwhichcanbe$$$$notatedasnorwhateveryoulike.$$

Commented by Tawa1 last updated on 30/Oct/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{I}\mathrm{understand}\mathrm{everything}\mathrm{now}.$$

Commented by MrW3 last updated on 31/Oct/18

$$nicetoknowthissir!$$$$justasanexercisepleasetry$$$$133x+65y-211z+47w=93$$

Commented by MrW3 last updated on 31/Oct/18

$$Ihaveputsomelargenumberssuchthat$$$$aparticularsolutioncannotbefound$$$$obviously.$$

Commented by Tawa1 last updated on 31/Oct/18

$$133\mathrm{x}+65\mathrm{y}+47\mathrm{w}-211\mathrm{z}=93$$$$133\mathrm{x}+65\mathrm{y}=\mathrm{u}......\mathrm{i}$$$$47\mathrm{w}-211\mathrm{z}=\mathrm{v}.....\mathrm{ii}$$$$\mathrm{u}+\mathrm{v}=93.......\mathrm{iii}$$$$$$$$\mathrm{From}\left(\mathrm{iii}\right)$$$$\mathrm{u}+\mathrm{v}=93$$$$\mathrm{u}=1+\mathrm{m},\mathrm{v}=92-\mathrm{m}$$$$$$$$\mathrm{from}\left(\mathrm{ii}\right)$$$$47\mathrm{w}-211\mathrm{z}=\mathrm{v}.....\mathrm{ii}$$$$\mathrm{w}=828-9\mathrm{m}-211\mathrm{n}$$$$\mathrm{z}=184-2\mathrm{m}-47\mathrm{n}$$$$$$$$\mathrm{from}\left(\mathrm{i}\right)$$$$133\mathrm{x}+65\mathrm{y}=\mathrm{u}......\mathrm{i}$$$$\mathrm{x}=22+22\mathrm{m}+65\mathrm{k}$$$$\mathrm{y}=-45-45\mathrm{m}-133\mathrm{k}$$$$$$$$\mathrm{Now}$$$$$$$$\mathrm{x}=22+22\mathrm{m}+65\mathrm{k}$$$$\mathrm{y}=-45-45\mathrm{m}-133\mathrm{k}$$$$\mathrm{z}=184-2\mathrm{m}-47\mathrm{n}$$$$\mathrm{w}=828-9\mathrm{m}-211\mathrm{n}$$$$$$

Commented by MrW3 last updated on 31/Oct/18

$$verygood!$$

Commented by Tawa1 last updated on 31/Oct/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Answered by MJS last updated on 29/Oct/18

$$7x+5y+3(5z+4w)=149$$$$5z+4w=v$$$$7x+5y+3v=149$$$$\mathrm{put}x=k$$$$5y=149-7k-3v$$$$\mathrm{put}v=ak+b\Rightarrow 5y=(149-3b)+(-7-3a)k$$$$\mathrm{put}a=1\Rightarrow 5\mid (-7-3a)k$$$$\mathrm{put}b=-2\Rightarrow 5\mid (149-3b)$$$$v=k-2$$$$\Rightarrow y=31-2k$$$$5z+4w=k-2$$$$5z=k-2-4w$$$$\mathrm{put}w=ak+b\Rightarrow 5z=(-2-4b)+(1-4a)k$$$$\mathrm{put}a=-1\Rightarrow 5\mid (1-4a)k$$$$\mathrm{put}b=2\Rightarrow 5\mid (-2-4b)$$$$w=-k+2$$$$z=k-2$$$$\mathrm{test}$$$$7k+5(31-2k)+15(k-2)+12(-k+2)=$$$$=7k+155-10k+15k-30-12k+24=$$$$=149$$$$$$$$\mathrm{general}\mathrm{solution}$$$$x=k$$$$y=31-2k$$$$z=k-2$$$$w=-k+2$$

Commented by MrW3 last updated on 29/Oct/18

$$thankyousir!$$$$butIthinkwhatyougotrepresentsonly$$$$apartofallpossiblesolutions.asa$$$$generalsolutionitshouldrepresent$$$$allpossiblesolutions.$$$$Forexample,$$$$x=92,y=243,z=-122,w=10$$$$isasolution,butit{doesn}^{\prime }tmatchwith$$$$thesolutionschemeyougave.$$$$Ithinkforanequationwith4unknowns$$$$weneedthreeparameterstodescribe$$$$thegeneralsolution.$$

Commented by MJS last updated on 29/Oct/18

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{right}.\mathrm{this}\mathrm{was}\mathrm{my}\mathrm{first}\mathrm{attempt}\mathrm{to}$$$$\mathrm{solve}\mathrm{an}\mathrm{equation}\mathrm{like}\mathrm{this}.\mathrm{now}\mathrm{I}\mathrm{think}\mathrm{we}\mathrm{can}$$$$\mathrm{freely}\mathrm{choose}2\mathrm{variables}\mathrm{because}\mathrm{the}\mathrm{other}$$$$2\left(\mathrm{if}\mathrm{factors}\mathrm{are}\mathrm{coprime}\right)\mathrm{will}\mathrm{do}\mathrm{all}\mathrm{the}\mathrm{work}$$$$\mathrm{for}\mathrm{us}$$$$\mathrm{in}\mathrm{this}\mathrm{case}\mathrm{choose}z\mathrm{and}w\mathrm{then}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{always}$$$$\mathrm{possible}\mathrm{to}\mathrm{find}\mathrm{a}\mathrm{matching}\mathrm{pair}(x,y)$$

Commented by MrW3 last updated on 29/Oct/18

$$thanksforreviewingsir!$$

Commented by MJS last updated on 29/Oct/18

$$...\mathrm{in}\mathrm{this}\mathrm{case}$$$$2a+3b+5c+7d=11$$$$\mathrm{we}\mathrm{can}\mathrm{choose}\mathrm{one}\mathrm{of}\mathrm{these}\mathrm{pairs}$$$$a/b,a/c,a/d,b/c,b/d,c/d$$$$\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{case}\mathrm{we}\mathrm{have}$$$$7x+5y+15z+12w$$$$\mathrm{we}\mathrm{can}\mathrm{only}\mathrm{choose}\mathrm{these}$$$$x/z,y/z,y/w,z/w$$$$\mathrm{does}\mathrm{this}\mathrm{change}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{parameters}?$$$${\mathrm{I}}^{\prime }\mathrm{m}\mathrm{a}\mathrm{bit}\mathrm{confused}...$$

Commented by Tawa1 last updated on 29/Oct/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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