x^2+y^2 = 25..(1)

x^3+y^3 = 91..(2)

To solve for x and y.

Solution:

By guess the solution x = 4 and y = 3. Or y = 3 and x = 4, as

x^2+y^2 = 4^2+3^2 = 25 and

x^3+y^3 = 4^3+3^3 = 64+27 = 91.

We knowthat x^3+y^3 = (x+y) (x^2+y^2 -xy).

Therefore, (x+y)(x^2+y^2-xy) = 91 for the given data.

(x+y)(x^2+y^2-xy) = 13*7. Or 7*13.

Let us check for x+y = 7 and x^2+y^2-xy = (x+y)^2-3xy = 7^2 -3xy = 13 . Or

49-3xy = 13.

3xy = 49-13 = 36

xy = 36/4 =12.

So x+y =7..........(1). Therefore,

x-y = sqrt{(x+y)^2 -4xy }

x-y = {49-48} = 1.

x-y = 1...............(2).

Add eq(1) and eq(2):

2x = 7+1=8

x= 8/2 = 4.

Eq(1)-eq(2) gives:

2y = 7-1 = 6.

y = 6/2 = 3.

Therefore (x , y) = (4,3). Also (x,y) = (3,4).

We'll solve the system, starting from the following formulas:

x^2 + y^2 = (x+y)^2 - 2xy

x^3 + y^3 = (x+y)^3 - 3xy(x+y)

Now, we'll note x+y = S and xy = P.

x^2 + y^2 = S^2 - 2P

x^3 + y^3 = S^3 - 3SP

We'll substitute them into the equations of the system:

S^2 - 2P = 25 (1)

S^3 - 3SP = 91 (2)

We'll factorize (2):

S(S^2 - 3P) = 91

We'll extract S^2 from (1)=>S^2 = 25+2P

S(25+2P - 3P) = 91

S(25-P) = 91

We'll remove the brackets:

25S - SP = 91 => SP = 25S - 91

S^3 - 3SP = 91

S^3 - 3 (25S - 91) = 91

S^3 - 75S = 91 - 3*91

S(S^2 - 75) = -2*91

S = -2

S^2 - 75 = 91

S^2 = 91+75

S^2 = 166