Question

Find the solutions of x and y in the following:x^3 + 9x^2y = 10 y^3 + xy^2 = 2

Answer 1

Given :   x³  + 9x²y = 10 , y³  + xy²  = 2

To find  :   x & y

Solution:

x³  + 9x²y = 10

=> x² (x + 9y) = 10

y³  + xy²  = 2

=> y²(x + y)  = 2

=> 5y²(x + y) = 10

x² (x + 9y) = 5y²(x + y)

Let say x = ky

=> (ky)²(ky + 9y) = 5y²(ky + y)

=> y³ k²(k + 9) = 5y³(k + 1)

=> k³ + 9k²   = 5k + 5

=> k³ + 9k²  - 5k - 5 = 0

=> (k - 1)(k² + 10k + 5) = 0

=> k = 1  or k  =  (-10  ±  √80)/2  = -5 ± 2√5  

(ignoring -ve values of k )

k = 1  => x = y

=> x³  + x(x)²  = 2  => 2x³ = 2  => x³ = 1

=> x  = 1

, y = 1

x = 1 , y = 1  

Verification :

1³ + 9(1)²1 = 10 => 10 = 10

1³ + 1(1)² = 2 => 2 = 2

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