Question

Система уравнений. 3^x*2^y=12 2^y+1-3^x=5

Answer 1

$\left \{ {{3^x*2^y=12 } \atop {2^{y+1}-3^x=5}} \right. \\2^y=u,\ u\ \textgreater \ 0 \\3^x=v,\ v\ \textgreater \ 0 \\ \left \{ {{v*u=12} \atop {2u-v=5}} \right. \\v=2u-5 \\(2u-5)*u=12 \\2u^2-5u-12=0 \\D=25+96=121=11^2 \\u_1= \frac{5-11}{4} \ \textless \ 0 \\u_2= \frac{5+11}{4} =4 \\v=2*4-5=3 \\2^y=4 \\2^y=2^2 \\y=2 \\3^x=3 \\x=1$
ответ: (1;2)

Answer 2

Сделаем замену
$3 ^{x}=a0,$    $2 ^{y}=b\ \textgreater \ 0$,тогда $2 ^{y+1} = 2b$
$\left \{ {{a*b=12} \atop {2b-a=5}} \right.\\\\ \left \{ {{a=2b-5} \atop {(2b-5)*b=12}} \right.\\\\ \left \{ {{a=2b-5} \atop {2b ^{2}-5b-12=0 }} \right.$
2b² - 5b - 12 = 0
D = 25 -4 * 2 * (- 12) = 25 + 96 = 121
$b _{1}= \frac{5+ \sqrt{121} }{4} = \frac{5+11}{4} = \frac{16}{4}=4\\\\b _{2}= \frac{5- \sqrt{121} }{4}= \frac{5-11}{4} = \frac{-6}{4} = - 1,5$ не уд
$\left \{ {{b=4} \atop {a=2*4-5}} \right.\\\\ \left \{ {{b=4} \atop {a=3}} \right. \\\\ \left \{ {{3 ^{x}=3 } \atop {2 ^{y} =4}} \right.\\\\ \left \{ {{x=1} \atop {y=2}} \right.$
ответ: (1 ; 2)