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Question Number 92448 by jagoll last updated on 07/May/20

$$\{\begin{array}{l}{5}^{\mathrm{x}}.6^{\mathrm{y}}=150\\ 5^{\mathrm{y}}.6^{\mathrm{x}}=180\end{array}$$

Commented by john santu last updated on 07/May/20

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Answered by john santu last updated on 07/May/20

Commented by jagoll last updated on 07/May/20

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Answered by $@ty@m123 last updated on 07/May/20

$$5^x.6^x.5^y.6^x=150\times 180$$$$5^{x+y}.6^{x+y}=27000$$$${30}^{x+y}={30}^3$$$$x+y=3$$

Commented by jagoll last updated on 07/May/20

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Answered by Rasheed.Sindhi last updated on 07/May/20

$$6(5^x.6^y)=6\left(150\right)$$$$\Rightarrow 5^x.6^{y+1}=900......\left(i\right)$$$$5(5^{\mathrm{y}}.6^{\mathrm{x}})=5\left(180\right)$$$$\Rightarrow 5^{y+1}.6^x=900.....\left(ii\right)$$$$\left(i\right)/\left(ii\right):(5^x.6^{y+1})/(5^{y+1}.6^x)=1$$$$5^{x-y-1}.6^{y-x+1}=5^0.6^0$$$$\Rightarrow x-y=1...............................\left(i\right)$$$$x+y=3\left(ByS@ty@msir\right)....\left(ii\right)$$$$From\left(i\right)\mathrm{\&}\left(ii\right)$$$$x=2,y=1$$$$$$

Answered by Rasheed.Sindhi last updated on 07/May/20

$$ByFactorization$$$$150and180canbewrittenas$$$$productof5\mathrm{\&}6:$$$$\{\begin{array}{l}{5}^x.6^y=150=2.3.5^2=5^2.6^1\\ 5^y.6^x=180=2^2.3^2.5=5^1.6^2\end{array}$$$$\Rightarrow x=2\wedge y=1$$

Commented by Prithwish Sen 1 last updated on 08/May/20

$$\mathrm{wow}!$$

Commented by Rasheed.Sindhi last updated on 08/May/20

$$\mathcal{T}h\alpha n\mathcal{X}sir.$$$$Moreover:$$$$Ifinanequation{a}^x.b^y=c,ccanbe$$$$writtenasaproductofa\mathrm{\&}bin$$$$a\mathit{u}\mathit{n}\mathit{i}\mathit{q}\mathit{u}\mathit{e}way\left(asinthiscase\right)only$$$$oneequationissufficient!$$

Commented by Prithwish Sen 1 last updated on 08/May/20

$$\mathrm{agree}$$

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