Point Satisfying Line Equation from Infinitely Many Solutions System

To find the condition for the given system of linear equations to have infinitely many solutions, we express the third equation as a linear combination of the first two equations. Let $\text{Equation 3} = \lambda (\text{Equation 1}) + \mu (\text{Equation 2})$

Comparing the coefficients of $x$ and $y$, we get the system: $\lambda + 2\mu = 1$ $5\lambda + 3\mu = 6$

Solving these equations simultaneously yields $\lambda = \frac{9}{7}$ and $\mu = -\frac{1}{7}$. Using these values to determine $a$ and $b$, we find: $a = 6\lambda + 4\mu = 6\left(\frac{9}{7}\right) + 4\left(-\frac{1}{7}\right) = \frac{50}{7}$ $b = 4\lambda + 7\mu = 4\left(\frac{9}{7}\right) + 7\left(-\frac{1}{7}\right) = \frac{29}{7}$

Thus, the point $(a, b) = \left(\frac{50}{7}, \frac{29}{7}\right)$ satisfies $a - b = \frac{50 - 29}{7} = 3$, which corresponds to the line equation $x - y = 3$.

Hence, the correct option is (B).