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Question Number 18388 by tawa tawa last updated on 19/Jul/17

$$\mathrm{Solve}\mathrm{simultaneously}.$$$$\mathrm{x}+\mathrm{y}=5.......\left(\mathrm{i}\right)$$$$5^{\mathrm{x}}+\mathrm{y}=15......\left(\mathrm{ii}\right)$$

Answered by mrW1 last updated on 19/Jul/17

$$5^{\mathrm{x}}-\mathrm{x}=10$$$$5^{\mathrm{x}}=\mathrm{x}+10$$$$5^{\mathrm{x}+10}=(\mathrm{x}+10)5^{10}$$$${\mathrm{e}}^{(\mathrm{x}+10)\ln 5}=(\mathrm{x}+10)5^{10}$$$$-(\mathrm{x}+10)\ln {5\mathrm{e}}^{-(\mathrm{x}+10)\ln 5}=-5^{-10}\ln 5$$$$-(\mathrm{x}+10)\ln 5=\mathrm{W}(-5^{-10}\ln 5)$$$$\Rightarrow \mathrm{x}=-\frac{\mathrm{W}(-5^{-10}\ln 5)}{\ln 5}-10\approx -10$$$$\Rightarrow \mathrm{y}=\frac{\mathrm{W}(-5^{-10}\ln 5)}{\ln 5}+15\approx 15$$

Commented by tawa tawa last updated on 19/Jul/17

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

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