positive number
Number Theory
Level
3
Let x,y and z be positive numbers
such that x+y+xy=8 y+z+yz=15 z+x+zx=35
find the value of x+y+z+xy
The answer is 15.
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1 solution
Rewrite\quad x+y+xy+1=9\quad etc,\quad and\quad factoring,\ \ \begin{align} (x+1)(y+1) & =9 \ (y+1)(z+1) & =16 \ (x+1)(z+1) & =36 \end{align}\ Their\quad product\quad is\ \ \begin{align} (x+1)^{ 2 }(y+1)^{ 2 }(z+1)^{ 2 }=9\cdot 16\cdot 36\quad \Longrightarrow \quad (x+1)(y+1)(z+1)=72{ 4 } \ \end{align}\ Divide\quad each\quad of(1),(2),(3)from(4)\quad to\quad get\quad z+1=8,x+1=4.5,y+1=2,and\quad so\quad x+y+z+xy=3.5+1+7+3.5=\boxed { 15 } .