Question

If (y + z)(z + x)(x + y) = $\sf\frac{45}{4}$, (x + 1)(y + 1)(z + 1) = 9 and xyz = 1, then the value of x + y + 2z is equal to? ​

Answer 1

||✪✪ GIVEN ✪✪||

• (y + z)(z + x)(x + y) = (45/4)
• (x + 1)(y + 1)(z + 1) = 9
• xyz = 1

✯✯ To Find ✯✯

• ( x + y + 2z) = ?

|| ✰✰ ANSWER ✰✰ ||

Let ,

→ (y + z)(z + x)(x + y) = (45/4) ------------- Equation ❶

→ (x + 1)(y + 1)(z + 1) = 9 --------------- Equation ❷

→ xyz = 1 --------------- Equation ❸

Also,

Let, (x + y + z) = k (Any Constant Term).

So,

→ (x + y) = (k - z)

→ (y + z) = (k - x)

→ (z + x) = (k - y)

Putting These Value in Equation ❶ Now, we get,

➼ (k - z)(k - x)(k - y) = (45/4)

➼ k³ - k²(x + y + z) + k(xy + yz + zx) - xyz = (45/4)

Putting value of xyz from Equation ❸ & (x+y+z) as k now,

➼ k³ - k³ + k(xy + yz + zx) - 1 = (45/4)

➼ k³ - k³ + k(xy + yz + zx) = (45/4) + 1

➼ k³ - k³ + k(xy + yz + zx) = (49/4) .

➼ k(xy + yz + zx) = (49/4) ------- Equation ⓸

__________________________

Now, Expanding Equation ❶ :-

➺ (x + 1)(y + 1)(z + 1) = 9

➺ (x + y + z) + (xy + yz + zx) + xyz + 1 = 9

Putting value of xyz from Equation ❸ & (x+y+z) as k again,

➺ k + (xy + yz + zx) + 2 = 9

➺ (xy + yz + zx) = 9 - 2 - k = (7 - k)

➺ (xy + yz + zx) = ( 7 - k)

Multiply both sides by k

➺ k(xy + yz + zx) = k( 7 - k)

Putting value of Equation ⓸ Here, now,

➺ (49/4) = 7k - k²

➺ 49 = 28k - 4k²

➺ 4k² - 28k + 49 = 0

➺ (2k)² - 2*2k*7 + (7)² = 0 {a² - 2ab + b² = (a - b)²}.

➺ (2k - 7)² = 0

➺ 2k - 7 = 0

➺ 2k = 7

➺ k = (7/2). ----------- Equation ❺

___________________

So,

➻ (x + y + z) = k = (7/2)

➻ (x + y) = (7/2 - z) --------------- Equation ➏

Now,

Putting value of k From Equation ❺ in Equation ❹ , we get,

➔ (7/2)(xy + yz + zx) = (49/4)

➔ (xy + yz + zx) = (49/4) * (2/7) = (7/2)

➔ z(x + y) + xy = (7/2)

Putting value of Equation ➏ & from ❸ { xy = 1/z } , we get,

➠ z (7/2 - z) + 1/z = (7/2)

➠ (7z/2 - z²/1) + (1/z) = (7/2)

Taking LCM,

➠ (7z² - 2z³ + 2) /2z = (7/2)

➠ (7z² - 2z³ + 2) = 7z

➠ 2z³ - 7z² + 7z - 2 = 0

Solving This we get,

☛ Z = 1.

___________________

Putting This value in Equation ➏ now, we get,

☛ (x + y) = (7/2 - 1)

☛ (x + y) = (5/2)

So ,

☛ (x + y + 2z)

☛(5/2) + 2*1

☛(5/2) + 2

☛ (9/2) (Ans).

Hence, The value of (x + y + 2z) will be (9/2).

______________________________

Answer 2

$\mathfrak{\huge{\red{ANSWER}}}$

In Attached File......