# Solve system u+v=2,ux+vy=3,ux^2+vy^2=5,ux^3+vy^3=9

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### 1 Answer

You need to solve the following system of equations, such that:

`{(u+v=2),(ux+vy=3),(ux^2+vy^2=5),(ux^3+vy^3=9):}`

You may use Lagrange's resolvents, such that:

`{(x + y = p),(xy = q):} => {(x^2 - px + q = 0),(y^2 - py + q = 0):}`

Replacing x and y by `x^2 - px + q = 0` and `y^2 - py + q = 0` , yields:

`u*(x^2 - px + q) + v*(y^2 - py + q) = 0`

`ux^2 - p*u*x + q*u + vy^2 - v*p*y + v*q = 0`

Grouping the terms yields:

`(ux^2 + vy^2) - s(ux + vy) + p(u + v) = 0`

You should replace 5 for `ux^2 + vy^2` (the third equation),3 for `ux + vy` (the second equation) and 2 for `u + v` (the top equation), such that:

`5 - 3s + 2p = 0 => -3s + 2p = -5`

You need to replace `x^2 - px + q` and `y^2 - py + q` for x and y in the third equation, such that:

`ux*(x^2 - px + q) + vy*(y^2 - py + q) = 0`

`(ux^3 + vy^3) - p(ux^2 + vy^2) + q(ux + vy) = 0`

You should replace 9 forĀ `ux^3 + vy^3` (the bottom equation), 5 for `ux^2 + vy^2` (the third equation),3 for ux + vy (the second equation), such that:

`9 - 5p + 3q = 0 => - 5p + 3q = -9`

You should solve the following system of equations, such that:

`{(-3s + 2p = -5),(- 5p + 3q = -9):} => {(9s - 6p = 15),(- 10p + 6q = -18):}`

Adding the equations, yields:

`-p = -3 => p = 3`

`-15 + 3q = -9 => 3q = 6 => q = 2`

Replacing 3 for `p` and 2 for `q` in equations `x^2 - px + q = 0` and `y^2 - py + q = 0` yields:

`x_(1,2) = y_(1,2) = (3+-sqrt(9 - 8))/2 => x_1 = y_1 = 1; x_2 = y_2 = 2`

You need to evaluate u and v, such that:

`{(u + v = 2),(u + 2v = 3):} => v = 1 => u = 1`

`{(u + v = 2),(2u + v = 3):} => v = 1 => u = 1`

**Hence, evaluating x,y,u and v, under the given conditions, yields `x = 1, y=2, x=2, y=1, u = v = 1` .**

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