Question

if 2x + 3y = 8 and xy=2 find the value of: a. 8x^3 + 27y^3 and b (2x-3y)^3

Answer 1

Answer:

the value of: a. 8x^3 + 27y^3 = 224 and b (2x-3y)^3 = 64

Step-by-step explanation:

Given , 2x + 3y = 8 & x × y = 2

∴ $(2x+3y)^{2}$ = $8^{2}$ = 64  → (a)

again , 2x 3y = 6xy = 12  →(b)

from (a) , (2x)^2 + (3y)^2 + 2 (2x) (3y) = 64

⇒ (2x)^2 + (3y)^2 + 2 × 12 = 64 [ from (b) ]

⇒ (2x)^2 + (3y)^2 = 64 - 24 = 40 → (c)

⇒  (2x)^2 + (3y)^2 - 2 (2x) (3y) = 40 - 24 = 16

⇒ $(2x-3y)^{2}$ = 16

⇒ 2x-3y = 4

⇒ $(2x-3y)^{3}$ = 64

now 8x^3 + 27y^3 = (2x)^3 + (3y)^3 = (2x + 3y) {(2x)^2 - (2x) (3y) +  (3y)^2 }

= 8 ×  {(2x)^2 +  (3y)^2 - (2x) (3y) }

= 8 × (40 - 12) [from (b) & (c)]

= 8 ×28 = 224