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Question Number 92013 by I want to learn more last updated on 04/May/20

$$\mathrm{Solve}:$$$$2^{\mathrm{x}}+3^{\mathrm{y}}=72.....\left(\mathrm{i}\right)$$$$2^{\mathrm{y}}+3^{\mathrm{x}}=108.....\left(\mathrm{ii}\right)$$

Commented by jagoll last updated on 04/May/20

$$108\times 2=216$$$$\mathrm{why}416??$$

Commented by jagoll last updated on 04/May/20

$$\mathrm{x}=\mathrm{y}+2$$$$2^{\mathrm{y}+2}+3^{\mathrm{y}}=72$$$$2^{\mathrm{y}}+3^{\mathrm{y}+2}=108$$$$\mathrm{let}{2}^{\mathrm{y}}=\mathrm{t}\wedge 3^{\mathrm{y}}=\mathrm{w}$$$$4\mathrm{t}+\mathrm{w}=72$$$$\mathrm{t}+9\mathrm{w}=108$$$$\mathrm{now}\mathrm{easy}\mathrm{to}\mathrm{solve}$$

Commented by I want to learn more last updated on 04/May/20

$$\mathrm{Thanks}\mathrm{sir},\mathrm{i}\mathrm{appreciate}.$$

Commented by I want to learn more last updated on 04/May/20

$$\mathrm{Why}\mathrm{do}\mathrm{we}\mathrm{let}\mathrm{x}=\mathrm{y}+2$$$$\mathrm{can}\mathrm{i}\mathrm{still}\mathrm{say}\mathrm{x}=\mathrm{y}+5$$

Commented by jagoll last updated on 04/May/20

$$\mathrm{eq}\left(1\right)\times \mathrm{by}3$$$$\mathrm{eq}\left(2\right)\times \mathrm{by}2$$$$\mathrm{substract}\left(1\right)\mathrm{\&}\left(2\right)$$

Commented by I want to learn more last updated on 04/May/20

$$3\left(2^{\mathrm{x}}\right)+3^{\mathrm{y}+1}=216$$$$2^{\mathrm{y}+1}+2\left(3^{\mathrm{x}}\right)=216$$$$\mathrm{Please}\mathrm{next}\mathrm{step}\mathrm{here}$$$$2^{\mathrm{y}+1}-3\left(2^{\mathrm{x}}\right)+2\left(3^{\mathrm{x}}\right)-3^{\mathrm{y}+1}=0$$$$2\left(3^{\mathrm{x}}\right)-3\left(2^{\mathrm{x}}\right)=3^{\mathrm{y}+1}-2^{\mathrm{y}+1}$$

Commented by I want to learn more last updated on 04/May/20

$$\mathrm{sir},\mathrm{show}\mathrm{the}\mathrm{steps}\mathrm{till}\mathrm{when}\mathrm{you}\mathrm{say}\mathrm{let}\mathrm{x}=\mathrm{y}+2.$$$$\mathrm{i}\mathrm{will}\mathrm{do}\mathrm{the}\mathrm{rest}$$

Commented by I want to learn more last updated on 04/May/20

$$\mathrm{Have}\mathrm{changed}\mathrm{it}.\mathrm{please}\mathrm{next}\mathrm{steps}.$$

Commented by I want to learn more last updated on 04/May/20

$$\mathrm{t}=15.429\mathrm{and}\mathrm{w}=10.286$$$$2^{\mathrm{y}}=15.429$$$$\mathrm{y}=\frac{\ln (15.429)}{\ln \left(2\right)}=\frac{2.736}{0.6931}=3.9475$$$$\mathrm{x}=3.9475+2=5.9475$$$$$$$$\mathrm{But}\mathrm{not}\mathrm{correct}\mathrm{with}\mathrm{the}\mathrm{question}$$

Commented by MJS last updated on 05/May/20

$$x=y+2\mathrm{is}\mathrm{wrong}$$$$\mathrm{the}\mathrm{path}\mathrm{given}\mathrm{to}\mathrm{get}x=y+2\mathrm{is}\mathrm{wrong}$$

Answered by MJS last updated on 04/May/20

$$\mathrm{approximating}\mathrm{leads}\mathrm{to}$$$$x\approx 4.15063$$$$y\approx 3.63495$$

Commented by I want to learn more last updated on 04/May/20

$$\mathrm{But}\mathrm{using}\mathrm{the}\mathrm{approach}\mathrm{above},\mathrm{am}\mathrm{not}\mathrm{getting}\mathrm{it}$$

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