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Solve-2-x-3-y-72-i-2-y-3-x-108-ii-

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Question Number 92013 by I want to learn more last updated on 04/May/20
Solve: 2^x + 3^y = 72 ..... (i) 2^y + 3^x = 108 ..... (ii)
$$\mathrm{Solve}: \\ $$$$\:\:\:\mathrm{2}^{\mathrm{x}} \:\:+\:\:\mathrm{3}^{\mathrm{y}} \:\:\:=\:\:\mathrm{72}\:\:\:\:\:…..\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\mathrm{2}^{\mathrm{y}} \:\:+\:\:\mathrm{3}^{\mathrm{x}} \:\:\:=\:\:\mathrm{108}\:\:\:\:\:…..\:\left(\mathrm{ii}\right) \\ $$
Commented by jagoll last updated on 04/May/20
108×2=216 why 416??
$$\mathrm{108}×\mathrm{2}=\mathrm{216} \\ $$$$\mathrm{why}\:\mathrm{416}?? \\ $$
Commented by jagoll last updated on 04/May/20
x=y+2 2^(y+2) +3^y = 72 2^y +3^(y+2) = 108 let 2^y =t ∧ 3^y =w 4t+w = 72 t+9w=108 now easy to solve
$$\mathrm{x}=\mathrm{y}+\mathrm{2}\: \\ $$$$\mathrm{2}^{\mathrm{y}+\mathrm{2}} +\mathrm{3}^{\mathrm{y}} \:=\:\mathrm{72} \\ $$$$\mathrm{2}^{\mathrm{y}} +\mathrm{3}^{\mathrm{y}+\mathrm{2}} \:=\:\mathrm{108}\: \\ $$$$\mathrm{let}\:\mathrm{2}^{\mathrm{y}} =\mathrm{t}\:\wedge\:\mathrm{3}^{\mathrm{y}} =\mathrm{w} \\ $$$$\mathrm{4t}+\mathrm{w}\:=\:\mathrm{72} \\ $$$$\mathrm{t}+\mathrm{9w}=\mathrm{108}\: \\ $$$$\mathrm{now}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{solve} \\ $$
Commented by I want to learn more last updated on 04/May/20
Thanks sir, i appreciate.
$$\mathrm{Thanks}\:\mathrm{sir},\:\mathrm{i}\:\mathrm{appreciate}. \\ $$
Commented by I want to learn more last updated on 04/May/20
Why do we let x = y + 2 can i still say x = y + 5
$$\mathrm{Why}\:\mathrm{do}\:\mathrm{we}\:\mathrm{let}\:\:\:\:\mathrm{x}\:\:=\:\:\mathrm{y}\:+\:\mathrm{2} \\ $$$$\mathrm{can}\:\mathrm{i}\:\mathrm{still}\:\mathrm{say}\:\:\:\:\:\mathrm{x}\:\:=\:\:\mathrm{y}\:\:+\:\:\mathrm{5}\: \\ $$
Commented by jagoll last updated on 04/May/20
eq(1) × by 3 eq(2) × by 2 substract (1) &(2)
$$\mathrm{eq}\left(\mathrm{1}\right)\:×\:\mathrm{by}\:\mathrm{3} \\ $$$$\mathrm{eq}\left(\mathrm{2}\right)\:×\:\mathrm{by}\:\mathrm{2}\: \\ $$$$\mathrm{substract}\:\left(\mathrm{1}\right)\:\&\left(\mathrm{2}\right) \\ $$
Commented by I want to learn more last updated on 04/May/20
3(2^x ) + 3^(y + 1) = 216 2^(y + 1) + 2(3^x ) = 216 Please next step here 2^(y + 1) − 3(2^x ) + 2(3^x ) − 3^(y + 1) = 0 2(3^x ) − 3(2^x ) = 3^(y + 1) − 2^(y + 1)
$$\:\:\:\:\:\:\mathrm{3}\left(\mathrm{2}^{\mathrm{x}} \right)\:\:+\:\:\mathrm{3}^{\mathrm{y}\:\:+\:\:\mathrm{1}} \:\:\:=\:\:\:\mathrm{216} \\ $$$$\:\:\:\:\:\mathrm{2}^{\mathrm{y}\:+\:\mathrm{1}} \:\:+\:\:\mathrm{2}\left(\mathrm{3}^{\mathrm{x}} \right)\:\:=\:\:\mathrm{216} \\ $$$$\mathrm{Please}\:\mathrm{next}\:\mathrm{step}\:\mathrm{here} \\ $$$$\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{y}\:+\:\mathrm{1}} \:\:−\:\:\mathrm{3}\left(\mathrm{2}^{\mathrm{x}} \right)\:\:\:+\:\:\mathrm{2}\left(\mathrm{3}^{\mathrm{x}} \right)\:\:−\:\:\mathrm{3}^{\mathrm{y}\:+\:\mathrm{1}} \:\:=\:\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\mathrm{2}\left(\mathrm{3}^{\mathrm{x}} \right)\:\:−\:\:\mathrm{3}\left(\mathrm{2}^{\mathrm{x}} \right)\:\:=\:\:\mathrm{3}^{\mathrm{y}\:+\:\mathrm{1}} \:\:−\:\:\mathrm{2}^{\mathrm{y}\:\:+\:\:\mathrm{1}} \\ $$
Commented by I want to learn more last updated on 04/May/20
sir, show the steps till when you say let x = y + 2. i will do the rest
$$\mathrm{sir},\:\mathrm{show}\:\mathrm{the}\:\mathrm{steps}\:\mathrm{till}\:\mathrm{when}\:\mathrm{you}\:\mathrm{say}\:\mathrm{let}\:\:\mathrm{x}\:=\:\mathrm{y}\:+\:\mathrm{2}. \\ $$$$\mathrm{i}\:\mathrm{will}\:\mathrm{do}\:\mathrm{the}\:\mathrm{rest} \\ $$
Commented by I want to learn more last updated on 04/May/20
Have changed it. please next steps.
$$\mathrm{Have}\:\mathrm{changed}\:\mathrm{it}.\:\mathrm{please}\:\mathrm{next}\:\mathrm{steps}. \\ $$
Commented by I want to learn more last updated on 04/May/20
t = 15.429 and w = 10.286 2^y = 15.429 y = ((ln(15.429))/(ln(2))) = ((2.736)/(0.6931)) = 3.9475 x = 3.9475 + 2 = 5.9475 But not correct with the question
$$\mathrm{t}\:\:=\:\:\mathrm{15}.\mathrm{429}\:\:\:\:\mathrm{and}\:\:\:\:\:\mathrm{w}\:\:\:=\:\:\:\mathrm{10}.\mathrm{286} \\ $$$$\:\:\:\:\:\mathrm{2}^{\mathrm{y}} \:\:=\:\:\mathrm{15}.\mathrm{429} \\ $$$$\:\:\:\:\:\:\mathrm{y}\:\:=\:\:\frac{\mathrm{ln}\left(\mathrm{15}.\mathrm{429}\right)}{\mathrm{ln}\left(\mathrm{2}\right)}\:\:\:=\:\:\:\frac{\mathrm{2}.\mathrm{736}}{\mathrm{0}.\mathrm{6931}}\:\:\:=\:\:\:\mathrm{3}.\mathrm{9475} \\ $$$$\:\:\:\:\mathrm{x}\:\:\:=\:\:\:\mathrm{3}.\mathrm{9475}\:\:+\:\:\mathrm{2}\:\:\:=\:\:\mathrm{5}.\mathrm{9475} \\ $$$$ \\ $$$$\mathrm{But}\:\mathrm{not}\:\mathrm{correct}\:\mathrm{with}\:\mathrm{the}\:\mathrm{question} \\ $$
Commented by MJS last updated on 05/May/20
x=y+2 is wrong the path given to get x=y+2 is wrong
$${x}={y}+\mathrm{2}\:\mathrm{is}\:\mathrm{wrong} \\ $$$$\mathrm{the}\:\mathrm{path}\:\mathrm{given}\:\mathrm{to}\:\mathrm{get}\:{x}={y}+\mathrm{2}\:\mathrm{is}\:\mathrm{wrong} \\ $$
Answered by MJS last updated on 04/May/20
approximating leads to x≈4.15063 y≈3.63495
$$\mathrm{approximating}\:\mathrm{leads}\:\mathrm{to} \\ $$$${x}\approx\mathrm{4}.\mathrm{15063} \\ $$$${y}\approx\mathrm{3}.\mathrm{63495} \\ $$
Commented by I want to learn more last updated on 04/May/20
But using the approach above, am not getting it
$$\mathrm{But}\:\mathrm{using}\:\mathrm{the}\:\mathrm{approach}\:\mathrm{above},\:\mathrm{am}\:\mathrm{not}\:\mathrm{getting}\:\mathrm{it} \\ $$

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