Solving System of Equations with Variables

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The discussion focuses on solving three systems of equations involving variables. The first system includes equations that can be manipulated using substitution, specifically by replacing variables like 3x with u and 3y with v. The second system involves roots and exponents, where one participant suggests solving for 3^x to simplify the process. The third system presents a more complex equation that may require careful manipulation of terms. Participants are seeking guidance on effective methods to approach these equations.
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1st: 32x-2y+2*3x-y-3=0
3x+31-y=4


2nd: 3y*\sqrt[x]{64}=36
5y*\sqrt[x]{512}=200

3rd: 9*5x+7*2x+y=457
6*5x-14*2x+2=-890

At first i treid to replace 3x with u , 3x=u and 3y=v but I don't know what to do then.

At 2nd, \sqrt[x]{64} I replace with 26/x but then this be more complicate,and I don't know another way,please help me!
 
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For 1,
Cant you solve the second equation for 3^x and then plug it back into the first one after you break it down using your exponent properties?

that's just suggestion at first glance.
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Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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