Consider the following system of equations:
$$x + 2y - 3z = a$$
$$2x + 6y - 11z = b$$
$$x - 2y + 7z = c$$
where $$a, b$$ and $$c$$ are real constants. Then the system of equations:

The system of equations is: $$x + 2y - 3z = a$$, $$2x + 6y - 11z = b$$, $$x - 2y + 7z = c$$.

We compute the determinant of the coefficient matrix: $$\Delta = \begin{vmatrix} 1 & 2 & -3 \\ 2 & 6 & -11 \\ 1 & -2 & 7 \end{vmatrix}$$.

Expanding: $$\Delta = 1(42 - 22) - 2(14 + 11) + (-3)(-4 - 6) = 20 - 50 + 30 = 0$$.

Since $$\Delta = 0$$, the system does not have a unique solution for all values of $$a, b, c$$. We check for consistency by performing row operations. Subtracting $$2R_1$$ from $$R_2$$: $$0x + 2y - 5z = b - 2a$$. Subtracting $$R_1$$ from $$R_3$$: $$0x - 4y + 10z = c - a$$.

Adding $$2 \times R_2'$$ to $$R_3'$$: $$0 = (c - a) + 2(b - 2a) = c + 2b - 5a$$.

For the system to be consistent, we need $$5a = 2b + c$$. When this condition holds, the system has a free variable (since rank is 2 with 3 unknowns), giving infinitely many solutions.

Therefore the system has an infinite number of solutions when $$5a = 2b + c$$.