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Question Number 95394 by I want to learn more last updated on 24/May/20

$$\mathrm{Solve}:\mathrm{x}+\mathrm{y}=3....\left(\mathrm{i}\right)$$$${\mathrm{x}}^{\mathrm{y}}+{\mathrm{y}}^{\mathrm{x}}=6.....\left(\mathrm{ii}\right)$$

Commented by mr W last updated on 25/May/20

$$nosolution.$$$$from\left(ii\right):$$$$x^y+y^x=6\Rightarrow x+y⩾\frac{2\ln 3}{W\left(\ln 3\right)}\approx 3.651$$$$butfrom\left(i\right):$$$$x+y=3$$

Commented by I want to learn more last updated on 25/May/20

$$\mathrm{Wow}\mathrm{please}\mathrm{sir}.$$$$\mathrm{What}\mathrm{of}$$$$\mathrm{x}+\mathrm{y}=5......\left(\mathrm{i}\right)$$$${\mathrm{x}}^{\mathrm{y}}+{\mathrm{y}}^{\mathrm{x}}=17......\left(\mathrm{ii}\right)$$$$\mathrm{x}=2\mathrm{and}\mathrm{y}=3.$$$$\mathrm{Please}\mathrm{sir}\mathrm{show}\mathrm{the}\mathrm{part}.\mathrm{Thanks}\mathrm{sir}.$$

Commented by mr W last updated on 25/May/20

$$x+y⩾\frac{2\ln \frac{17}{2}}{W\left(\frac{17}{2}\right)}\approx 4.841$$$$withx+y=5>4.841therearetwo$$$$solutions:x=2,y=3orx=3,y=2.$$$$inthiscasethesolutionsareobvious.$$$$wecannotcalculatethemusinga$$$$formula.forexampleforx+y=6$$$$thesolutionsarenolongerobvious.$$$$youcangetthemonlyapproximately.$$

Commented by I want to learn more last updated on 25/May/20

$$\mathrm{Thanks}\mathrm{sir}.\mathrm{I}\mathrm{appreciate}.$$

Commented by I want to learn more last updated on 25/May/20

$$\mathrm{How}\mathrm{do}\mathrm{we}\mathrm{separate}\mathrm{x}+\mathrm{y}\mathrm{to}$$$$\mathrm{get}\mathrm{x}+\mathrm{y}⩾\frac{2\ln \left(\frac{17}{2}\right)}{\mathrm{W}\left(\frac{17}{2}\right)}\mathrm{sir}$$

Commented by I want to learn more last updated on 25/May/20

$$\mathrm{Because}\mathrm{i}\mathrm{see}\mathrm{questions}\mathrm{like}:$$$${\mathrm{x}}^{\mathrm{x}}+{\mathrm{y}}^{\mathrm{y}}=31.....\left(\mathrm{i}\right)$$$$\mathrm{x}+\mathrm{y}=5....\left(\mathrm{ii}\right)$$$$\mathrm{So},\mathrm{i}\mathrm{was}\mathrm{wondering}\mathrm{if}\mathrm{any}\mathrm{method}\mathrm{could}\mathrm{solve}\mathrm{it}.$$$$\mathrm{x}=2\mathrm{and}\mathrm{y}=3$$

Commented by mr W last updated on 25/May/20

$${let}^{\prime }slookat{x}^y+y^x=awitha>0$$$$thiscurveissymmetricabouty=x.$$$${let}^{\prime }slookatx+y=b,{it}^{\prime }saline$$$$perpendiculartoy=x.$$$$seepicture:$$

Commented by mr W last updated on 25/May/20

Commented by I want to learn more last updated on 25/May/20

$$\mathrm{Thanks}\mathrm{sir}.\mathrm{I}\mathrm{appreciate}$$

Commented by mr W last updated on 25/May/20

$$wefindalinex+y=kwhichtangents$$$$thecurve{x}^y+y^x=aatpointA.$$$$ifx+y=b<k\Rightarrow nosolution.$$$$ifx+y=b=k\Rightarrow onesolution.$$$$ifx+y=b>k\Rightarrow twosolutions.$$$$$$$$saypointA(p,p),wehave$$$$p^p+p^p=a$$$$p^p=\frac{a}{2}$$$$\frac{\ln \left(\frac{a}{2}\right)}{p}{e}^{\frac{\ln \left(\frac{a}{2}\right)}{p}}=\ln \left(\frac{a}{2}\right)$$$$\Rightarrow \frac{\ln \left(\frac{a}{2}\right)}{p}=W\left(\ln \frac{a}{2}\right)$$$$\Rightarrow p=\frac{\ln \frac{a}{2}}{W\left(\ln \frac{a}{2}\right)}$$$$linex+y=kpassesthroughA(p,p)$$$$\Rightarrow k=2p=\frac{2\ln \frac{a}{2}}{W\left(\ln \frac{a}{2}\right)}$$

Commented by I want to learn more last updated on 12/Dec/20

$$\mathrm{Sir},\mathrm{for}{\mathrm{x}}^{\mathrm{x}}+{\mathrm{y}}^{\mathrm{y}}=\mathrm{a}\mathrm{and}\mathrm{x}+\mathrm{y}=\mathrm{b},$$$$\mathrm{Please}\mathrm{how}\mathrm{can}\mathrm{we}\mathrm{know}\mathrm{it}\mathrm{is}\mathrm{solvable}\mathrm{just}\mathrm{like}\mathrm{the}\mathrm{above}:{\mathrm{x}}^{\mathrm{y}}+{\mathrm{y}}^{\mathrm{x}}=\mathrm{a},\mathrm{x}+\mathrm{y}=\mathrm{b}.$$

Commented by I want to learn more last updated on 12/Dec/20

$$\mathrm{Sir}\mathrm{what}\mathrm{of}{\mathrm{x}}^{\mathrm{x}}+{\mathrm{y}}^{\mathrm{y}}=\mathrm{a},\mathrm{and}\mathrm{x}+\mathrm{y}=\mathrm{b}.$$$$\mathrm{How}\mathrm{can}\mathrm{we}\mathrm{know}\mathrm{it}\mathrm{is}\mathrm{solvable}\mathrm{like}\mathrm{the}\mathrm{one}\mathrm{you}\mathrm{showed}\mathrm{above}.$$

Commented by mr W last updated on 12/Dec/20

Commented by I want to learn more last updated on 12/Dec/20

Commented by I want to learn more last updated on 12/Dec/20

$$\mathrm{Sir},\mathrm{i}\mathrm{mean}\mathrm{is}\mathrm{it}\mathrm{still}\mathrm{the}\mathrm{same}\mathrm{as}\mathrm{this}.$$

Commented by mr W last updated on 12/Dec/20

$$youcanproceedinsimilaryway,but$$$$notexactlythesameway.$$

Commented by I want to learn more last updated on 13/Dec/20

$$\mathrm{Thanks}\mathrm{sir}.$$

Commented by I want to learn more last updated on 13/Dec/20

$$\mathrm{Sir},\mathrm{i}\mathrm{will}\mathrm{really}\mathrm{love}\mathrm{it}\mathrm{if}\mathrm{you}\mathrm{can}\mathrm{show}\mathrm{me}\mathrm{sir}.$$$$\mathrm{Am}\mathrm{sorry}\mathrm{to}\mathrm{be}\mathrm{taking}\mathrm{your}\mathrm{time}.\mathrm{Please}\mathrm{help}\mathrm{me}\mathrm{sir}.$$

Commented by Tawa11 last updated on 19/Jul/21

$$\mathrm{Sir}\mathrm{mrW},\mathrm{i}\mathrm{searched}\mathrm{old}\mathrm{post}.\mathrm{Please}\mathrm{show}\mathrm{this}\mathrm{sir}.\mathrm{It}\mathrm{is}\mathrm{similar}\mathrm{to}$$$$\mathrm{my}\mathrm{question}:{\mathrm{x}}^{\mathrm{x}}+{\mathrm{y}}^{\mathrm{y}}=13,\mathrm{x}+\mathrm{y}=5.$$$$\mathrm{I}\mathrm{will}\mathrm{love}\mathrm{to}\mathrm{just}\mathrm{know}\mathrm{the}\mathrm{conditions}\mathrm{if}\mathrm{the}\mathrm{question}\mathrm{is}\mathrm{solvable}\mathrm{or}\mathrm{not}.$$$$\mathrm{Just}\mathrm{as}\mathrm{you}\mathrm{showed}\mathrm{above}.\mathrm{Thanks}\mathrm{sir}.$$

Commented by mr W last updated on 20/Jul/21

$$ihavealreadysaid.$$

Commented by mr W last updated on 20/Jul/21

$$youcanusethemethodaboveand$$$$youshouldgetthat{x}^x+y^y=13and$$$$x+y=5havenosolution.$$

Commented by Tawa11 last updated on 20/Jul/21

$$\mathrm{I}\mathrm{will}\mathrm{try}\mathrm{sir}.$$

Commented by Tawa11 last updated on 20/Jul/21

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

Commented by Tawa11 last updated on 25/Jul/21

$$\mathrm{Sir},\mathrm{please}\mathrm{help}\mathrm{me}.\mathrm{I}\mathrm{am}\mathrm{not}\mathrm{getting}\mathrm{it}.\mathrm{Please}.$$

Commented by Tawa11 last updated on 25/Jul/21

$$\mathrm{Please}!!!$$

Commented by mr W last updated on 25/Jul/21

$$ihavesaidallwhaticansayandwhat$$$$iknow.youknowthegraphof$$$$x^y+y^x=aandthatfor{x}^x+y^y=aand$$$$youknowwhenthelinex+y=b$$$$intersectsthesegraphesandthen$$$$getifthereisasolutionortwo$$$$solutionsornosolution.iputall$$$$thisinfollowingdiagram:$$

Commented by mr W last updated on 25/Jul/21

Commented by mr W last updated on 25/Jul/21

$$youhavea=13,$$$$k=\frac{2\ln \frac{a}{2}}{W\left(\ln \frac{a}{2}\right)}=\frac{2\ln \frac{13}{2}}{W\left(\ln \frac{13}{2}\right)}=4.552$$$$x+y=5>4.552$$$$youcangetfromthediagramthat$$$$x^x+y^y=13andx+y=5havenosolution.$$

Commented by mr W last updated on 26/Jul/21

$$forthesamereasonyoucangetthat$$$$x^y+y^x=13andx+y=5havetwosolutions.$$$$butthereisnowaytocalculatethe$$$$exactsolutions.theycanonlybe$$$$approximated.$$

Commented by Tawa11 last updated on 26/Jul/21

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{I}\mathrm{really}\mathrm{appreciate}\mathrm{your}\mathrm{time}.\mathrm{Long}\mathrm{life}\mathrm{and}\mathrm{good}\mathrm{health}$$$$\mathrm{and}\mathrm{wealth}.$$

Commented by Tawa11 last updated on 09/Aug/21

$$\mathrm{Sir},\mathrm{I}\mathrm{have}\mathrm{solved}:{\mathrm{x}}^{\mathrm{y}}+{\mathrm{y}}^{\mathrm{x}}=13\mathrm{and}\mathrm{x}+\mathrm{y}=5.$$$$\mathrm{I}\mathrm{got}\mathrm{x}\approx 1.672\mathrm{and}\mathrm{y}\approx 3.328$$

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