Solve-x-2-y-2-xy-19-x-y-5-

Question Number 218036 by dscm last updated on 26/Mar/25

S o l v e

x^{2} + y^{2} + xy = 19

x + y = 5

Answered by Hanuda354 last updated on 26/Mar/25

x^{2} + y^{2} + xy = 19

{(x + y)}^{2} - x y = 19

5^{2} - x y = 19

25 - x y = 19

\Leftrightarrow x y = 6

Then we get :

(x, y) = (2, 3), (3, 2)

Answered by dscm last updated on 26/Mar/25

x^{2} + y^{2} + xy = 19 \dots . (i)

x + y = 5 \dots \dots (i i)

(i i) \Rightarrow x^{2} + y^{2} + 2 xy = 25 \dots . (i i i)

(i i i) - (i) : xy = 6

L e t x & y a r e t h e r o o t s o f a q u a d r a t i c e q u a t i o n

∵ x + y = 5 & xy = 6

∴ T h e e q u a t i o n m a y b e t^{2} - (x + y) t + x y = 0

O r t^{2} - 5 t + 6 = 0 \Rightarrow t = 2, 3

∴ (x = 2 \land y = 3) \lor (x = 3 \land y = 2)

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