I want to determine for which $\lambda\in\mathbb{R}$ the following linear system over $\mathbb{R}$ has a solution:

\begin{cases}
x\_1 + 2x\_2 - 3x\_3 = \lambda\\
3x\_1 + 5x\_2 - 3x\_3 = 2\lambda\\
-3x\_1-4x\_2-3x\_3=1
\end{cases}

I would like to understand how to solve this exercise. Here's the augmented matrix to make it a little easier to solve:

\begin{bmatrix}
1&2&-3&\lambda\\
3&5&-3&2\lambda\\
-3&-4&-3&1
\end{bmatrix}

I know that systems like these have a solution (one or infinite) if the rank of the coefficients matrix A matches the rank of the augmented matrix $A|b$. However, I don't know how I would go about evaluating the rank of $A|b$. Sure, I could try to calculate the determinant of all the submatrices of order 3 until I find that one has determinant $\neq 0$ but that would take a while.

Also, I've tried to reduce the matrix in row-echelon form but I can't see how to go from there.

\begin{bmatrix}
1&2&-3&\lambda\\
0&-1&0&-\lambda\\
0&1&-6&3\lambda
\end{bmatrix}

Any hints?