We are given:
1) x + y = 5
2) x^2 + y^2 = 13
We need to find x^3 + y^3.
Step 1: Find the value of xy.
We know that (x + y)^2 = x^2 + 2xy + y^2.
Substitute the given values into this identity:
5^2 = 13 + 2xy
25 = 13 + 2xy
2xy = 25 - 13
2xy = 12
xy = 6.
Step 2: Use the sum of cubes formula.
The sum of cubes formula is x^3 + y^3 = (x + y)(x^2 - xy + y^2).
Substitute the known values into this formula:
x^3 + y^3 = (5)(13 - 6)
x^3 + y^3 = (5)(7)
x^3 + y^3 = 35.
Therefore, x^3 + y^3 = 35.