An algebra problem by Mohammad Khaza

Algebra Level 3

If x 2 = 3 x + 6 y x^{2} = 3x+6y and x y = 5 x + 4 y xy = 5x + 4y are satisfied, then which of the following are possible solutions to the pairs ( x , y ) (x,y) ?

(0,0), (6,4) (5,1), (0,0) (0,0), (9,9) (2,1), (7,3)

Mohammad Khaza by Mohammad Khaza , 20, Bangladesh

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1 solution

Tom Engelsman
Jul 7, 2017

Let y = x 2 − 3 x 6 y = \frac{x^2 - 3x}{6} from the first equation and substitute this expression into the second equation to obtain:

y = 5 x x − 4 = x 2 − 3 x 6 ⇒ x 3 − 7 x 2 − 18 x = 0 ⇒ x ( x + 2 ) ( x − 9 ) = 0 ⇒ x = − 2 , 0 , 9. y = \frac{5x}{x-4} = \frac{x^2 - 3x}{6} \Rightarrow x^3 - 7x^2 - 18x = 0 \Rightarrow x(x+2)(x-9) = 0 \Rightarrow x = -2,0,9.

These values in turn produce the final ordered-pairs of solutions: ( x , y ) = ( 0 , 0 ) ; ( 9 , 9 ) ; ( − 2 , 5 3 ) (x,y) = \boxed{(0,0); (9,9); (-2,\frac{5}{3})} .

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