If `b+c = 2x, c+a =2y and a +b =2z` then value of `a^(3) +b^(3) +c^(3)` is

Options

A. `(x+y +z)^(3) `
B. `(x+y+z)^(3) +24xyz`
C. `(x+y+z)^(3) -24xyz`
D. 24xyz

Answer

To find the value of \( a^3 + b^3 + c^3 \) given the equations \( b + c = 2x \), \( c + a = 2y \), and \( a + b = 2z \), we can follow these steps:

### Step 1: Rewrite the equations
We have the following equations:
1. \( b + c = 2x \) (Equation 1)
2. \( c + a = 2y \) (Equation 2)
3. \( a + b = 2z \) (Equation 3)

### Step 2: Express variables in terms of others
From Equation 1, we can express \( b \) in terms of \( c \):
\[ b = 2x - c \]

### Step 3: Substitute \( b \) into Equation 3
Substituting \( b \) into Equation 3:
\[ a + (2x - c) = 2z \]
This simplifies to:
\[ a - c = 2z - 2x \quad \text{(Equation 4)} \]

### Step 4: Express \( a \) in terms of \( c \)
From Equation 2, we can express \( a \) in terms of \( c \):
\[ a = 2y - c \]

### Step 5: Substitute \( a \) into Equation 4
Substituting \( a \) into Equation 4:
\[ (2y - c) - c = 2z - 2x \]
This simplifies to:
\[ 2y - 2c = 2z - 2x \]
Dividing by 2 gives:
\[ y - c = z - x \]
Thus, we can express \( c \) as:
\[ c = y - z + x \]

### Step 6: Substitute \( c \) back to find \( a \) and \( b \)
Now substituting \( c \) back into the expression for \( a \):
\[ a = 2y - (y - z + x) = y + z - x \]

Now substituting \( c \) back into the expression for \( b \):
\[ b = 2x - (y - z + x) = x + z - y \]

### Step 7: Now we have expressions for \( a \), \( b \), and \( c \)
- \( a = y + z - x \)
- \( b = x + z - y \)
- \( c = y - z + x \)

### Step 8: Use the identity for \( a^3 + b^3 + c^3 \)
We can use the identity:
\[ a^3 + b^3 + c^3 = (a + b + c)^3 - 3(a + b)(b + c)(c + a) \]

### Step 9: Calculate \( a + b + c \)
\[ a + b + c = (y + z - x) + (x + z - y) + (y - z + x) = 2x + 2y + 2z - (x + y + z) = x + y + z \]

### Step 10: Calculate \( (a + b + c)^3 \)
\[ (a + b + c)^3 = (x + y + z)^3 \]

### Step 11: Calculate \( (a + b)(b + c)(c + a) \)
Using the original equations:
\[ (a + b) = 2z, \quad (b + c) = 2x, \quad (c + a) = 2y \]
Thus,
\[ (a + b)(b + c)(c + a) = (2z)(2x)(2y) = 8xyz \]

### Step 12: Substitute into the identity
Substituting back into the identity gives:
\[ a^3 + b^3 + c^3 = (x + y + z)^3 - 3(8xyz) = (x + y + z)^3 - 24xyz \]

### Final Answer
Thus, the value of \( a^3 + b^3 + c^3 \) is:
\[ \boxed{(x + y + z)^3 - 24xyz} \]