Question

Решить систему
X^2+y^2=37
Xy=6

Answer 1

${x}^{2} + {y}^{2} = 37 \ xy = 6 \ \ {x}^{2} + {y}^{2} + 2xy - 2xy = 37 \ xy = 6 \ \ {(x + y)}^{2} - 2xy = 37 \ xy = 6 \ \ {(x + y)}^{2} - 2 times 6 = 37 \ xy = 6 \ \ {(x + y)}^{2} = 49 \ xy = 6 \ \ 1)x + y = 7 \ xy = 6 \ \ y = 7 - x \ x(7 - x) = 6 \ \ y = 7 - x \ 7x - {x}^{2} = 6 \ \ y = 7 - x \ {x}^{2} - 7x + 6 = 0 \ \ x1 = 6 \ y1 = 1 \ \ x2 = 1 \ y2 = 6 \ \ 2) x + y = - 7 \ xy = 6 \ \ y = - 7 - x \ x( - 7 - x) = 6 \ \ y = - 7 - x \ - 7x - {x}^{2} = 6 \ \ y = - 7 - x \ {x }^{2} + 7x + 6 = 0 \ \ x1 = - 6 \ y1 = - 1 \ \ x2 = - 1 \ y2 = - 6$
Ответ: (6; 1), (1; 6), (-6; -1), (-1; -6)

Answer 2

$tt +left{begin{array}{I} tt x^2+y^2=37 \tt xy=6 | cdot 2end{array}}$

$tt x^2+2xy+y^2=49\ (x+y)^2=49\ x+y= pm 7\ \ left[begin{array}{I} left{begin{array}{I}tt x+y=7 \ tt xy=6 end{array}} \ left{begin{array}{I}tt x+y=-7 \ tt xy=6 end{array}} end{array}} Leftrightarrow left[begin{array}{I} tt (x; y)= (1; 6), (6; 1) \ tt (x; y)= (-6; -1), (-1; -6) end{array}}$

Ответ: (6; 1), (1; 6), (-6; -1), (-1; -6)