Решение
2^(2x - 2y) + 2^(x - y) = 2
2^(2x + 1) + (1/2)^(2y - 1) = 5
2^(2x - 2y) + 2^(x - y = 2
2^[2*(x - y)] + 2^(x - y) = 2
2^(x - y) = z, z > 0
z² + z - 2 = 0
z₁ = - 2 не удовлетворяет условию z > 0
z₂ = 1
2^(x - y) = 1
2^(x - y) = 2⁰
x - y = 0
x = y
подставим у = х во второе уравнение
2^(2x + 1) + (1/2)^(2y - 1) = 5
2^(2x + 1) + (1/2)^(2x - 1) = 5
2*2^(2x) + 2*(1/2)^(2x) - 5 = 0
умножим на 2^(2x)
2*2^(4x) - 5*2^(2x) + 2 = 0
2^2x = t
2t² - 5t + 2 = 0
D = 25 - 4*2*2 = 9
t₁ = (5 - 3)4
t₁ = 1/2
t₂ = (5 + 3)/4
t₂ = 2
1) 2^2x = 1/2
2^2x = 2⁻¹
2x = - 1
x₁ = - 1/2
2^2x = 2¹
2x = 1
x₂ = 1/2
так как x = y, то
y₁ = - 1/2
y₂ = 1/2
Ответ: (- 1/2; - 1/2) (1/2 ; 1/2)