System of 4 linear equations with 4 variables and two parameters

Question

I am trying to solve this system of equations with two parameters (through Gaussian elimination) and I have almost reached a solution but the numbers I get are terrible so I'd like to ask if I am on the right path to solving the problem. Unfortunately, my LaTex code in Word doesn't work if I paste it here, so I will share my solution through the photo below (please excuse me for the inconvenience:

Answer 1

If $\mu\ne 1$ the system has one unique solution, for Cramer's theorem $$\begin{cases} x_1=\frac{\lambda \mu +2 \lambda +5 \mu -8}{\mu -1}\\ x_2=\frac{9 \lambda \mu +8 \lambda +38 \mu -55}{\mu -1}\\ x_3=\frac{\lambda-1 }{\mu -1}\\ x_4=-\frac{4 \lambda \mu +4 \lambda +17 \mu -25}{\mu -1}\\ \end{cases}$$ if $\mu=1$ and $\lambda\ne 1$ the rank of the matrix of the coefficients is $3$ and the rank of the augmented matrix is $4$, therefore for Rouchè-Capelli theorem , as they are different, the system has no solutions.

If $\mu=1$ and $\lambda= 1$ the ranks are the same and are $3$ so we have infinite solutions depending on $4-3=1$ parameter.

$$ \begin{cases} x_1= \frac{3}{8} \left(x_4+5\right)\\ x_2= \frac{1}{8} \left(19-17 x_4\right)\\ x_3= -\frac{1}{8} \left(x_4+21\right)\\ \end{cases} $$