System with homogeneous equation in denominator help

AI Thread Summary
The discussion revolves around solving a system of equations with homogeneous denominators. The user attempts to simplify the equations by substituting 'x' with 'k*y', leading to complex expressions. Suggestions include substituting the terms 2x-y and x+2y with new parameters to eliminate denominators. A potential solution involves manipulating the first equation to express one variable in terms of another, allowing for substitution into the second equation. The conversation highlights the challenges of simplification and the need for careful algebraic manipulation.
Hivoyer
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Homework Statement


The system is declared as follows:

8/(2*x - y) - 7/(x + 2*y) = 1
4/((2*x - y)^2) - 7/((x + 2*y)^2) = 3/28

Homework Equations





The Attempt at a Solution



I define 'x' to equal k*y and I replace it inside the equation:

8/(2*k*y^2) - 7(k*y + 2*y) = 1
4(4*k^2*y^2 - 4*k*y^2 + y*2) - 7(k^2*y^2 + 4*k*y^2 + 4*y^2) = 3/28

I know I have to divide the top one by the bottom one and take y^2 out of the brackets, however the way it is now, it would become impossibly complex if I divide them.How can I simplify them before I divide?Not sure how to proceed.Using least common multiple also results in a monstrosity.
 
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How do you get 2ky^2 if you replace x with ky in 2x - y?
What happens to the fractions in equation 2?

I know I have to divide the top one by the bottom one
I doubt that this will work.

I would substitute 2x-y and x+2y with new parameters, and get rid of the denominators quickly.
 
mfb said:
How do you get 2ky^2 if you replace x with ky in 2x - y?
What happens to the fractions in equation 2?

I doubt that this will work.

I would substitute 2x-y and x+2y with new parameters, and get rid of the denominators quickly.

By doing that I get:
8*v - 7*u = u*v
12*v^2 - 196*u^2 = 3*u^2*v^2

I can't use any of them to express u = # or v = # :(
 
You can solve the first equation for v or u and plug it into the second equation.
 
Hivoyer said:
By doing that I get:
8*v - 7*u = u*v
12*v^2 - 196*u^2 = 3*u^2*v^2

I can't use any of them to express u = # or v = # :(

First, shouldn't the ##12v^2## in your second equation be ##112v^2##? (Probably just a typo)

Second, I suggest manipulating the first equation to get ##u = \frac{8v}{v+7}##(assuming that ##v \neq -7##, you should check that this cannot be true by substituting ##v = -7## into the system). Then, as mentioned above, substitute this expression into the second equation and solve for v.
 
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