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Determine the number of solutions of the simultaneous equations $x^2+y^3=29$ and $\log_3x\log_2y=1$.

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- Feb 14, 2012

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Determine the number of solutions of the simultaneous equations $x^2+y^3=29$ and $\log_3x\log_2y=1$.

Determine the number of solutions of the simultaneous equations $x^2+y^3=29$ and $\log_3x\log_2y=1$.

$\log_2 y = b \Rightarrow y = 2^b $

$a.b =1 $

$(3^a)^2 + (2^b)^3 = 29 $

$(3^{2/b}) + (2^{3b}) = 29 $

$f(x) = 3^{2/x} + 2^{3x} - 29 $

$f(1) = 9 + 8 - 29 < 0 $

$f(2) = 3 + 64 - 29 > 0 $ we have a zero at (1,2) interval

$f(1/2) = 3^4 + 2^{3/2} - 29 > 0 $ we have anther zero at (1/2 , 1 )

numbers larger than 2 , $2^{3x}$ is more than 29 so there is not any zero (2, infinity)

for x in (0,1/2) $3^{2/x}$ is more than 29 f(x) is positive

for negative numbers $3^{2/x} + 2^{3x} $ less than 29

so I think we have just two zeros

two solutions

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- Feb 14, 2012

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Hi

$\log_2 y = b \Rightarrow y = 2^b $

$a.b =1 $

$(3^a)^2 + (2^b)^3 = 29 $

$(3^{2/b}) + (2^{3b}) = 29 $

$f(x) = 3^{2/x} + 2^{3x} - 29 $

$f(1) = 9 + 8 - 29 < 0 $

$f(2) = 3 + 64 - 29 > 0 $ we have a zero at (1,2) interval

$f(1/2) = 3^4 + 2^{3/2} - 29 > 0 $ we have anther zero at (1/2 , 1 )

numbers larger than 2 , $2^{3x}$ is more than 29 so there is not any zero (2, infinity)

for x in (0,1/2) $3^{2/x}$ is more than 29 f(x) is positive

for negative numbers $3^{2/x} + 2^{3x} $ less than 29

so I think we have just two zeros

two solutions