MHB Determine the number of solutions for a system of equation

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The simultaneous equations $x^2+y^3=29$ and $\log_3x\log_2y=1$ can be analyzed to determine the number of solutions. The first equation describes a curve, while the second imposes a logarithmic constraint on the variables. Substitution methods can effectively simplify the analysis of these equations. The discussion confirms that the approach taken yields correct results, emphasizing the importance of substitution in solving such systems. Ultimately, the number of solutions hinges on the intersection of the two curves defined by the equations.
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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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anemone said:
Determine the number of solutions of the simultaneous equations $x^2+y^3=29$ and $\log_3x\log_2y=1$.

$\log_3 x = a \Rightarrow x = 3^a $
$\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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Amer said:
$\log_3 x = a \Rightarrow x = 3^a $
$\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

Hi Amer, thanks for participating and yes, your answer is correct and your method by using the substitution skill seems awesome too!
 
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