Question

If $x,y,z$ $ϵ$ $Q$ such that $(2x-y)+(2x-y)$ $\sqrt{3}$ $=(x+y)+(x-y-1)$ $\sqrt{5}$, then find $x,y,z$.

Answer 1

Taking rational numbers on left side and irrational number on right side, we get

$2x-y-(x+y)=(x-y-1)\sqrt{5}-(2x-y)\sqrt{3}=0$

$\Rightarrow x-2y=0$

$\Rightarrow x=2y$ .....(1)

$\Rightarrow (2y-y-1)\sqrt{5}-(4y-y)\sqrt{3}=0$

$\Rightarrow (y-1)\sqrt{5}=3\sqrt{3}y$, which is not possible because $y$ and $y-1$ are rational numbers,

Therefore, $y\in \varphi ⟹x\in \varphi$