## Linear Algebra and Its Applications (5th Edition)

$$\frac{c}{d}\neq\frac{1}{3}$$
The system is consistent for all values of $f$ and $g$, so it cannot be inconsistent for any values of $f$ and $g$. For the system to be inconsistent, it would require the left hand side of the second equation to be a scalar multiple of the left hand side of the first, with different right hand values. The condition $\frac{c}{d}\neq\frac{1}{3}$ ensures that the above cannot occur, ensuring consistency.