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Question Number 18388 by tawa tawa last updated on 19/Jul/17
Solve simultaneously.   x + y = 5     ....... (i)  5^x  + y = 15    ...... (ii)
$$\mathrm{Solve}\:\mathrm{simultaneously}.\: \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:=\:\mathrm{5}\:\:\:\:\:…….\:\left(\mathrm{i}\right) \\ $$$$\mathrm{5}^{\mathrm{x}} \:+\:\mathrm{y}\:=\:\mathrm{15}\:\:\:\:……\:\left(\mathrm{ii}\right) \\ $$
Answered by mrW1 last updated on 19/Jul/17
5^x −x=10  5^x =x+10  5^(x+10) =(x+10)5^(10)   e^((x+10)ln 5) =(x+10)5^(10)   −(x+10)ln 5e^(−(x+10)ln 5) =−5^(−10)  ln 5  −(x+10)ln 5=W(−5^(−10)  ln 5)  ⇒x=−((W(−5^(−10)  ln 5))/(ln 5))−10≈−10  ⇒y=((W(−5^(−10)  ln 5))/(ln 5))+15≈15
$$\mathrm{5}^{\mathrm{x}} −\mathrm{x}=\mathrm{10} \\ $$$$\mathrm{5}^{\mathrm{x}} =\mathrm{x}+\mathrm{10} \\ $$$$\mathrm{5}^{\mathrm{x}+\mathrm{10}} =\left(\mathrm{x}+\mathrm{10}\right)\mathrm{5}^{\mathrm{10}} \\ $$$$\mathrm{e}^{\left(\mathrm{x}+\mathrm{10}\right)\mathrm{ln}\:\mathrm{5}} =\left(\mathrm{x}+\mathrm{10}\right)\mathrm{5}^{\mathrm{10}} \\ $$$$−\left(\mathrm{x}+\mathrm{10}\right)\mathrm{ln}\:\mathrm{5e}^{−\left(\mathrm{x}+\mathrm{10}\right)\mathrm{ln}\:\mathrm{5}} =−\mathrm{5}^{−\mathrm{10}} \:\mathrm{ln}\:\mathrm{5} \\ $$$$−\left(\mathrm{x}+\mathrm{10}\right)\mathrm{ln}\:\mathrm{5}=\mathrm{W}\left(−\mathrm{5}^{−\mathrm{10}} \:\mathrm{ln}\:\mathrm{5}\right) \\ $$$$\Rightarrow\mathrm{x}=−\frac{\mathrm{W}\left(−\mathrm{5}^{−\mathrm{10}} \:\mathrm{ln}\:\mathrm{5}\right)}{\mathrm{ln}\:\mathrm{5}}−\mathrm{10}\approx−\mathrm{10} \\ $$$$\Rightarrow\mathrm{y}=\frac{\mathrm{W}\left(−\mathrm{5}^{−\mathrm{10}} \:\mathrm{ln}\:\mathrm{5}\right)}{\mathrm{ln}\:\mathrm{5}}+\mathrm{15}\approx\mathrm{15} \\ $$
Commented by tawa tawa last updated on 19/Jul/17
God bless you sir.
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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