$$x + 2y + z = 5(x + y)(y + z)\\ x + y + 2z = 7(y + z)(z + x)\\ 2x + y + z = 6(z + x)(x + y).$$ Find the value of $24^3xyz$.

This seems a problem concerning simultaneous equations. I didn't even know where to start! Please Help!

** EDIT** Can the OP confirm that it is $24^3 xyz$ and not $24^{3xyz}$? I am asking here to make it easier for OP to answer without using LaTeX.

  • $\begingroup$ @user44197 I'd bet the house that the OP wants $24^3 xyz$. $\endgroup$
    – David H
    Jan 5 '14 at 7:55
  • $\begingroup$ @user44197. Solving using your nice approach, it is effectively (24^3) x y z. $\endgroup$ Jan 5 '14 at 8:04
  • $\begingroup$ @DavidH. I agree with you. $\endgroup$ Jan 5 '14 at 8:05

Hint:Assume $x+y = a ,y+z = b, z+x = c$ and express both LHS and RHS of all 3 equations in terms of $a,b,c$. After some simplification, you should be able to get a system of linear equations in $\frac{1}{a},\frac{1}{b},\frac{1}{c}$.

Can you solve it now?


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