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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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