solve for value of X and Y if X2 +Y2=25 and X3+Y3=91?

Because we are having x^2+y^2 and x^3 + y^3 we can choose

(x+y) = a and xy = b

We get x^2+ y^2 =a^2 -2b = 25 ..1

And (x^3+y^3) = (x+y)^3 – 3xy(x+y)

or a^3 – 3ab = 91 ..2

so 2 (a^3 -3ab) – 3a(a^2- 2b) = 182-75a

or a^3 -75a + 182 = 0

this is cubic in a and by taking factors of 182(1,2,7,13,14,26,91,182) and –ve of them

we can see that a= 7 is a root and from (1) b = xy = 12

now x+ y = 7 and xy = 12 => (x-y)^2 = (x+y)^2 – 4xy = 1

so x- y = +/- 1

x-y = 1 => x= 4 and y = 3

x-y - = - 1 => x = 3 ,y = 4

so solution = (4,3) and (3,4)