1.
{x+y =π/2; sinx*cosy =1/2.⇔{x+y =π/2; (sin(x+y)+sin(x-y))/2 =1/2. ⇔
{x+y =π/2; (sin(π/2)+sin(x-y)) =1. ⇔ {x+y =π/2; 1+sin(x-y) =1.⇔
{x+y =π/2 ; x-y =π*n , n∈Z.⇒{x=π/4+πn/2 , y=π/4-πn/2 , n∈Z.
Ответ: x=π/4+πn/2 , y=π/4-πn/2 , n∈Z.
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2.
{lg(x-5) +√(y-3) =1;lg(x-5) -√(y-3) = -1. ||+/-||⇔{2lg(x-5) =0 ;2√(y-3) =2.
⇒{x-5=1 ;y -3 =1⇔{x=6 ;y =4.
Ответ: x =6 ;y =4.
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3.
{ 0,4^(1+log_0,4 (x-2y) =3,6; log_3 x -2log_0,3 y =-1.||ОДЗ: x>0;y>0||⇔
{ 0,4* (0,4^(log_0,4 (x-2y)) =3,6; log_3 x +log_3 3 =2log_0,3 y.⇔
{ x-2y =9; 3x = y².⇔ { y²/3-2y =9; x = y²/3.⇔ { y²-6y -27=0; x = y²/3.
⇒{y =9 ;(y =-3 не решение) ; x = 9²/3=27.
Ответ: x =27 ;y =9.
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4.
{∛x -∛y =1; 3∛x² -2∛xy =11 +∛y² * * * замена ∛x= u ;∛y=v * * *
{u -v =1; 3u² -2uv - v² =11.⇔{u=v+1; 3(v+1)² -2(v+1)v - v² =11.⇔
{u=v+1; 3v²+6v+3 -2v²-2v - v² =11.⇔{u=v+1; 4v =8.
⇔{u=3; v =2. ⇔{ ∛x =3 ;∛y =2 . ⇔{ x =27 ;y =8 .
Ответ: x =27 ; y =8.