MHB Solving System of Equations: xy, yz, zx

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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.
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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
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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