We know that X and y are positive numbers, and prove that: (x + y) (x ^ + y ^) (x ^ 3 + y ^ 3) ≥ 8x ^ 3Y ^ 3

We know that X and y are positive numbers, and prove that: (x + y) (x ^ + y ^) (x ^ 3 + y ^ 3) ≥ 8x ^ 3Y ^ 3

x. Y are all positive numbers, so
x+y≥2(xy)^(1/2)
x^2+y^2≥2xy
x^3+y^3≥2(xy)^(3/2)
Triple multiplication
Then: (x + y) (x ^ + y ^) (x ^ 3 + y ^ 3) ≥ 8x ^ 3Y ^ 3

8x-3y=11,x-y=-8
It takes a process to solve this equation,

8x-3y=11 (1)
x-y=-8 (2)
(1)-(2)×3
8x-3y-3x+3y=11+24
5x=35
x=7
y=x+8=15

Proof: if 4x-y is a multiple of 7, where x and y are integers, then 8x ^ 2 + 10xy-3y ^ 2 is a multiple of 49. Is that wrong?

If 4x-y is a multiple of 7, then 4x-y = 7m. Then y = 4x-7m
2x+3y = 2x+3*(4x-7m ) = 14x-21m =7 (2x-3m) .
8x ^ 2 + 10xy-3y ^ 2 = (4x-y) * (2x + 3Y) = 7m * 7 (2x-3m) = 49m (2x-3m), which is a multiple of 49