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:

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

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

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

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

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)