MHB Solving System of Equations: xy, yz, zx

AI Thread Summary
The discussion focuses on solving a system of equations involving three variables x, y, and z, defined by the equations xy/(x+y) = a, yz/(y+z) = b, and zx/(z+x) = c, with a, b, and c being non-zero constants. Participants explore various algebraic techniques and substitutions to simplify the equations and find relationships between the variables. The conversation emphasizes the importance of understanding the implications of the equations and potential methods for isolating variables. Solutions may involve manipulating the equations to express one variable in terms of the others or using numerical methods for specific values of a, b, and c. The thread highlights the complexity of the problem and the collaborative effort to derive a comprehensive solution.
solakis1
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Solve the following systemof equations:

$\dfrac{xy}{x+y}=a$

$\dfrac{yz}{y+z}=b$

$\dfrac{zx}{z+x}=c$

where a,b,c are not zero
 
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solakis said:
Solve the following systemof equations:

$\dfrac{xy}{x+y}=a$

$\dfrac{yz}{y+z}=b$

$\dfrac{zx}{z+x}=c$

where a,b,c are not zero
Inverting the 3 we get$\frac{1}{y} + \frac{1}{x} = \frac{1}{a}\dots(1)$$\frac{1}{y} + \frac{1}{z} = \frac{1}{b}\dots(2)$$\frac{1}{z} + \frac{1}{x} = \frac{1}{c}\dots(3)$Add the 3 to get$2(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$Or $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{2}(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$Subtracting (1) from above we get$\frac{1}{z} = \frac{1}{2}(\frac{1}{b} + \frac{1}{c}- \frac{1}{a})$Or $\frac{1}{z} = \frac{1}{2}\frac{ac + ab - bc}{abc}$Or $z= \frac{2abc}{ac + ab - bc}$ Similarly $x= \frac{2abc}{ab + bc - ac}$And $y= \frac{2abc}{ac - ab + bc}$
 
nery good
 
solakis said:
very good
 

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