Consider x + y + z = 6, x + 2y + 3z = 10 and x + 2y + λ z = µ…

Step 1. The determinant. $

= \lambda - 3$. A unique solution needs$\lambda \ne 3$.

Step 2. The case $\lambda = 3$. The last two equations are $x + 2y + 3z = 10$ and $x + 2y + 3z = \mu$.

Step 3. The value of $\mu$. If $\mu = 10$ they are the same equation, and there are infinitely many solutions. If $\mu \ne 10$ they contradict each other, and there is no solution.