system of linear equations

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Luke53
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system of linear equations

Is there an easy way to solve a system like this one?
x y/(3x-4y)=2/11
y z/(2y+3z)=6/5
x z/(x-z)=3/2
Luke

stapel_eliz
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x y/(3x-4y)=2/11
y z/(2y+3z)=6/5
x z/(x-z)=3/2
Since this is not a linear system, any solution method will likely be messy.

What have you tried so far?

Luke53
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Re: system of linear equations

Giving another example, of how it should be done according to my textbook:
given the following system:
x y/(x+y)=8/3
y z/(y+z)=8/5
x z/(x+z)=4/3
the first eqn is the same as:
(1/x)+(1/y)=3/8 (1)
the second eqn is:
(1/y)+(1/z)=5/8 (2)
the third eqn:
(1/x)+(1/z)=6/8 (3)
So the sum of the three last equations is equal to: 2/x+2/y+2/z=14/8; dividing this by 2 gives:
1/x+1/y+1/z=7/8 (4)
given the eqn's (2) and (4): 1/x= (7/8)-(5/8)= 2/8 and so x=4
out of (3) and (4): 1/y=7/8-6/8= 1/8 so y=8
and at last 1/z= (7/8)-(3/8)=4/8 and z=2 (out of (1 and (4))
I can't see how this can be applied to the eqn's given in the first system that I asked for an easier solution (and doesn't seem to be a linear system).
Luke.

nona.m.nona
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Re: system of linear equations

the first eqn is the same as:
(1/x)+(1/y)=3/8 (1)
the second eqn is:
(1/y)+(1/z)=5/8 (2)
the third eqn:
(1/x)+(1/z)=6/8 (3)
These are the same, assuming that none of x, y, and z is zero.

For convenience, rename so 8/x = X, 8/y = Y, and 8/z = Z. This gives you the following:

$\begin{array}{cccccc}X&+&Y&\,&=&3\\\,&\,Y&+&Z&=&5\\X&&+&Z&=&6\end{array}$

This is a linear system which can be solved. Then you will need to back-solve to find the values of x, y, and z.

Luke53
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Re: system of linear equations

Is there an easy way to solve a system like this one?
x y/(3x-4y)=2/11
y z/(2y+3z)=6/5
x z/(x-z)=3/2
Luke
Can one transform the three equations into the 1/x + 1/y form like in the previous example?
Applying the tranformation to the third eqn: x z/(x-z) = 3/2, would be something like: (1/z) - (1/x) = 2/3
How about transforming the first two equations into this form?

Greetings;
Luke.

nona.m.nona
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Re: system of linear equations

Can one transform the three equations into the 1/x + 1/y form like in the previous example?
The same process would likely work.

Luke53
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Re: system of linear equations

Luke.

stapel_eliz
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What part of the previous worked example do you not understand?

Luke53
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Re: system of linear equations

I know that : x y /(x+y) = 8/3 is the same as 1/x + 1/y = 3/8
Could this sort of transformation also be done with: x y / (3x - 4y) = 2/11 ? ( guess not, but I'm not sure).
Thanks.

stapel_eliz
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I know that : x y /(x+y) = 8/3 is the same as 1/x + 1/y = 3/8
Could this sort of transformation also be done with: x y / (3x - 4y) = 2/11 ?
As mentioned previously, "The same process would likely work." That means that the same sort of transformation likely can also be done with the posted system. So try it, and see what happens!