For the system of linear equations
`x+y+z=6`
`alpha x+beta y+7z=3`
`x+2y+3z=14`
Which of the following is NOT true?

Options

A. If `alpha=beta` and `alpha!=7,` then the system has a unique solution
B. There is a unique point `(alpha,beta)` on the line `x+2y+18=0` for which the system has infinitely many solutions
C. For every point `(alpha,beta)!=(7,7)` on the line `x-2y+7=0`,the system has infinitely many solutions
D. If `alpha=beta=7,` then the system has no solution

Answer

To solve the system of linear equations and determine which statement is NOT true, we will analyze the equations step by step.

The given equations are:
1. \( x + y + z = 6 \) (Equation 1)
2. \( \alpha x + \beta y + 7z = 3 \) (Equation 2)
3. \( x + 2y + 3z = 14 \) (Equation 3)

### Step 1: Express variables in terms of z
From Equation 1, we can express \( y \) in terms of \( z \):
\[ y = 6 - x - z \]

Now, substitute \( y \) into Equation 3:
\[ x + 2(6 - x - z) + 3z = 14 \]
Expanding this gives:
\[ x + 12 - 2x - 2z + 3z = 14 \]
Combining like terms:
\[ -x + z + 12 = 14 \]
This simplifies to:
\[ -x + z = 2 \quad \Rightarrow \quad x = z - 2 \quad \text{(Equation 4)} \]

### Step 2: Substitute into Equation 2
Now substitute \( x \) and \( y \) from Equations 4 and 1 into Equation 2:
\[ \alpha(z - 2) + \beta(6 - (z - 2) - z) + 7z = 3 \]
Simplifying \( y \):
\[ y = 6 - (z - 2) - z = 8 - 2z \]
Now substitute \( x \) and \( y \):
\[ \alpha(z - 2) + \beta(8 - 2z) + 7z = 3 \]
Expanding this gives:
\[ \alpha z - 2\alpha + 8\beta - 2\beta z + 7z = 3 \]
Combining like terms:
\[ (\alpha - 2\beta + 7)z + (8\beta - 2\alpha) = 3 \]

### Step 3: Analyze conditions for solutions
For the system to have:
- A unique solution: The coefficient of \( z \) must not equal zero:
\[ \alpha - 2\beta + 7 \neq 0 \]
- No solution: The coefficient of \( z \) must equal zero and the constant term must not equal zero:
\[ \alpha - 2\beta + 7 = 0 \quad \text{and} \quad 8\beta - 2\alpha \neq 3 \]
- Infinitely many solutions: Both the coefficient of \( z \) and the constant term must equal zero:
\[ \alpha - 2\beta + 7 = 0 \quad \text{and} \quad 8\beta - 2\alpha = 3 \]

### Conclusion
Now we can analyze the options given in the question to determine which statement is NOT true based on our derived conditions.