Решите систему уравнений.(номер 10)7 и 14 номер.Все досконально пишите.

7.

x^4 + x^2 y^2 + y^2 = 91

x^2 - xy + y^2 = 7

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x^4 + x^2 y^2 + y^4 = x^4 + 2 x^2 y^2 + y^4 - x^2 y^2 = (x^2 + y^2)^2 - (xy)^2 =

= (x^2 - xy + y^2)(x^2 + xy + y^2) = 7(x^2 + xy + y^2)

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x^2 + xy + y^2 = 91/7 = 13

x^2 - xy + y^2 = 7

(x^2 + xy + y^2) - (x^2 - xy + y^2) = 2xy = 13 - 7 = 6

xy = 3

x^2 + y^2 = 13 - 3 = 7 + 3 = 10

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x^2 + 2xy + y^2 = 10 + 6 = 16 = (x + y)^2

x^2 - 2xy + y^2 = 10 - 6 = 4 = (x - y)^2

x + y = 4

x - y = 2

При x gt;= y,

x - y = 2, y = x - 2

x + y = 2x - 2 = 4

2x - 2 = +-4

x = 1 +- 2, y = x-2 = -1 +- 2

При y gt;= x,

y = 1 +- 2, x = y-2 = -1 +- 2

Ответ: x = -1, y = -3 U x = 3, y = 1 U x = -3, y = -1 U x = 1, y = 3

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14.

x + y = 3z

x^2 + y^2 = 5z

x^3 + y^3 = 9z

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x^3 + y^3 = (x+y)(x^2 - xy + y^2) = 3z*(5z - xy) = 9z

5z - xy = 3

xy = 5z - 3

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5z = x^2 + y^2 = x^2 + 2xy + y^2 - 2xy = (x+y)^2 - 2(5z - 3) = 9z^2 - 10z + 6

9z^2 - 15z + 6 = 0

3z^2 - 5z + 2 = 0

(3z - 2)(z - 1) = 0

z1 = 2/3, z2 = 1

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x + y = 3z

x^2 + y^2 = x^2 - 2xy + y^2 + 2xy = (x-y)^2 + 2(5z-3) = 5z

(x-y)^2 = 6-5z

x-y = (6-5z)

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При z = 2/3,

x + y = 3z = 2

x-y = (6 - 10/3) = (8/3) = 2(2/3)

При x gt; y, x = 1 + (2/3), y = 1 - (2/3)

При y gt; x, y = 1 + (2/3), x = 1 - (2/3)

При z = 1,

x + y = 3z = 3

x - y = (6-5) = 1

При x gt; y, x = 2, y = 1

При y gt; x, y = 2, x = 1

Ответ: x = 1, y =2, z = 1 U x = 2, y =1, z = 1 U

U x = 1 - (2/3), y = 1 + (2/3), z = 2/3 U x = 1 +(2/3), y = 1 - (2/3), z = 2/3