Question

Решить систему
$\left \{ {{7^x-5^\frac{y}{2} =1776} \atop {7^\frac{x}{2}-5^\frac{y}{4} =24}} \right.$

Answer 1

Замена переменной:

$7^{x}=u\\\\7^{\frac{x}{2}}=(7^{x})^{\frac{1}{2}}=u^{\frac{1}{2}}=\sqrt{u}$

$5^{\frac{y}{2}}=v\\\\5^{\frac{y}{4}}=(5^{\frac{y}{2}})^{\frac{1}{2}}=v^{\frac{1}{2}}=\sqrt{v}$

$\left \{ {{u-v=1776} \atop {\sqrt{u}-\sqrt{v}=24 }} \right. \\\\\left \{ {{(\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})=1776} \atop {\sqrt{u}-\sqrt{v}=24 }} \right. \\\\\left \{ {{24\cdot (\sqrt{u}+\sqrt{v})=1776} \atop {\sqrt{u}-\sqrt{v}=24 }} \right.$

$\\\\\left \{ {\sqrt{u}+\sqrt{v}=74} \atop {\sqrt{u}-\sqrt{v}=24 }} \right.$

складываем:

2√u=98

√u=49

u=49²

u=7⁴

7ˣ=7⁴

x=4

√v=25

v=25⁴

$5^{\frac{y}{2}}=5^4\\\\\frac{y}{2}=4\\\\y=8$

О т в е т. (4;8)