1. Write the system in matrix form

The equations are

Coefficient matrix:

For a unique solution, we need $\det(A)\neq 0$. For infinitely many or no solution, we need $\det(A)=0$ and then check consistency.

2. Compute the determinant

Expanding along the first row:

So,

Hence:

• if $\alpha\neq 7$, the system has a unique solution;
• if $\alpha=7$, then we must check consistency.

3. Case $\alpha=7$

Then the system becomes

Subtract $7\times$ first equation from second:

Also, subtract $3\times$ first equation from third:

Using $y=4$ in $(\beta-7)y=-39$:

Therefore when $\alpha=7$:

• if $\beta=-\frac{11}{4}$, the system is consistent and since $\det(A)=0$, it has infinitely many solutions;
• if $\beta\neq -\frac{11}{4}$, the system is inconsistent, so no solution.

4. Check each option

Option A

If $\alpha=\beta=7$, then with $\alpha=7$ and $\beta=7\neq -\frac{11}{4}$, the system has no solution.

So A is true.

Option B

Line:

Here this means

The system has infinitely many solutions only when

Check whether this point lies on the line:

So the point of infinite solutions is not on this line.

Also, for any point on this line with $\alpha\neq 7$, determinant is nonzero, so solution is unique, not infinite.

Hence B is false.

Option C

Line:

meaning

For infinitely many solutions we need

Check if it lies on the line:

So there is no such point on this line.

Thus C is false.

Option D

If $\alpha=\beta$ and $\alpha\neq 7$, then certainly $\alpha\neq 7$, so

Hence the system has a unique solution.

So D is true.

5. Conclusion

The statements that are NOT true are:

So this question, as written, has two incorrect options, not one.