Question

find x and y if xsquare+ysquare=13andxcube+y cube=35.

Answer 1

x^2 + y^2 = 13
x^3 + y^3 = 35
by the you can solve this with logically
Let x = 2 and y = 3
2^2 + 3^2 = 4 + 9 = 13
2^3 + 3^3 = 8 + 27 = 35
hence, x = 2 and y = 3

process :---------------------------
x^2 + y^2 = 13
(x + y)^2 -2xy = 13___________(1)
again,
x^3 + y^3 = 35
(x + y)^3 -3xy (x + y) = 35__________(2)

Let (x + y) = P and xy = Q

eqns convert in
P^2 -2Q = 13
Q =( P^2 -13)/2-----------(3)
put it below eqn
P^3 -3PQ = 35
P^3 - 3 (P^2 - 13)/2 .P = 35
2P^3 - 3P^3 +39P = 70
P^3 - 39P + 70 = 0
P^3 - 5P^2 + 5P^2 -25P -14P + 70 =0
(P-5)(P^2 +5P -14) = 0
(P-5)(P^2 + 7P-2P-14)= 0
(P -5)(P-2)(P+7) = 0
P = 2, 5 and -7
but P doesn't possible 2 and -7 because x^3 + y^3 > 0 .
Q = ( 5^2 -13)/2 = 6

x + y = 5 and xy = 6
we observed x = 2 and y = 3