Алгебра, вопрос задал alexs12525 , 7 лет назад

решить систему уравнений
9^x -2^y =1
{
9^-x -2^-y=-1/6

Ответы на вопрос

Ответил sangers1959

Ответ: x=0,5 y=1.

Объяснениe:

$left { {{9^{x}-2^{y} =1} atop {frac{1}{9^{x} } -frac{1}{2^{y} } =-frac{1}{6} }} right.$

Пусть:

$9^{x}=u>0;2^{y}=v>0.\left { {{u-v=1} atop {frac{1}{u}-frac{1}{v} } =-frac{1}{6} }} right. ;left { {{u-v=1} atop {frac{v-u}{u*v} =-frac{1}{6} }} right. ;left { {{u-v=1} atop {\frac{u-v}{u*v} =frac{1}{6} }} ;right. ;left { {{u=v+1} atop {frac{1}{u*v} =frac{1}{6} }} right. ;left { {{u=v+1} atop {u*v=6}} ;right.\left { {{u=v+1} atop {v*(v+1)=6}} right. ;left { {{u=v+1} atop {v^{2}+v-6 =0}} right. ;left { {{u=v+1} atop {v^{2}+v+2v-2v-6 =0}} right. ;$

$left { {{u=v+1} atop {v^{2} +3v-2v-6=0}} right. ;left { {{u=v+1} atop {v*(v+3)-2*(v+3)=0}} right. ;left { {{u=v+1} atop {(v+3)*(v-2)=0}} right. ;left { {{u_{1}=-2 ;u_{2} =3} atop {v_{1} =-3;v_{2} =2}} right.$

u₁=-2 ∉; v₁=-3 ∉ ⇒

$left { {{u=9^{x} =3} atop {v^{2} =2^{y} =2}} right. ;left { {{3^{2x} =3^{1} } atop {2^{y} =2^{1} }} right. ;left { {2x=1|:2} atop {y=1}} right. ;left { {{x=0,5} atop {y=1}} right. .$