On several occasions I've seen involving Gaussian elimination to solve a system of equations, while this method doesn't seem to add anyting to the process, and the system must be solved using regular algebraic inference.
E.g. from example 4.11.12 here:
As you can see there is no additional explanation about how Gaussian elimination is further conducted. My understanding is Gaussian elimination is not used at all, the solution $t \, (1, -2, 1)$ is just found from the equation system:
$$ \begin{cases} x_{1}-x_{3} & =0\\ x_{2}+2x_{3} & =0 \end{cases} \Rightarrow \begin{cases} x_{1} & =x_{3}\\ x_{2} & =-2x_{3} \end{cases} \Rightarrow (x_{1,}x_{2,}x_{3})=t\,(1,-2,1) $$
But I'm not sure. What does the textbook wanted to say by "using Gaussian elimination" in this example?
For most 2x2 or 2x3 systems of equations, simple algebra is the way to go. But for larger systems, GE is a very helpful methodical approach to solving them. Book-authors often want to get you used to the idea that when you see a linear system, you bring out the standard tool (GE) to solve it. This isn't a great example, I admit, but writing a 4 x 6 example would have taken a lot of space, and all the steps of GE would have occupied more, etc.
One thing you get from GE is that at some point you may have an all-zero row in the main block, but a nonzero number in the far right; if that happens, you know your equations are inconsistent and there's no solution and you can stop working.
Another is that when you're done with GE, on the left you have a row-reduced matrix and perhaps $k$ of the rows have leading 1's, and $n-k$ do not. If the right-hand side of your system was all zeros, this tells you that you have an $n-k$-dimensional space of solutions to your system; if the right-hand side was nonzero, then it tells you that if there is any solution, then there is an $n-k$-parameter family of solutions. Those observations do not always instantly arise when you have 7 equations in 11 variables scribbled all over your page.