The following augmented matrix represents a equation system.
$$\left[\begin{array}{ccc|c} 1 & 2 & 4 & 8 \\ 0 & 1 & 4 & 9 \\ 0 & 0 & 1 & 6 \end{array}\right] $$
Since the matrix is in a row-Echelon form, it is possible to calculate the values of $x_1, x_2 ~and~ x3$ straightforward.
$$ \begin{array}\\ x_1 + 2x_2 + 4 x_3 = 8 \\ x_2 + 4x_3 = 9 \\ x_3 = 6 \end{array} $$
One of the steps describes the calculation of $x_2$. I need to describe that step as part of the solution to the equation system and I am not sure if the following representation is usually used to represent it:
$$x_2 + 4x_3 = 9 \color{red}{\wedge} x_3 = 6 \color{red}\to x_2 = -15$$
Just to clarify: the question is about semantics.
Technically, your notation is correct since you are expressing (as Lee Mosher said in a comment) "... and ... implies" using the standard boolean operation symbols from propositional logic. But a beginner student who reads that notation probably will not realize that it is in fact insufficient for the desired purpose of solving the original system of equations.
Take any real $x≥-2$. The following statements are true:
This does not mean that the following is true:
In general, if you want to solve a system of simultaneous equations, you generally perform some reasoning under the assumption that all those equations hold, and so the conclusions you reach under that assumption will restrict the possible values that can satisfy those equations. In other words, what you actually have would be that if $x,...,y$ satisfy some equations, then $x,...,y$ satisfy some constraints, where the constraints are simpler than the original equations. But as the above example shows, an assignment of values for $x,...,y$ that satisfies the constraints may not satisfy the original equations, so you still need to check which assignments do.
It turns out that for systems of linear equations, you can perform a special kind of reasoning where you transform the entire system through a sequence of equivalent systems. It is wrong to think of solving a linear equation as a series of steps where each step is simply obtaining a new equation from previous ones, such as the step in your question. If you are careful, you can think in terms of changing one part of the system at a time. Consider the following equivalence ("iff" means "if and only if") for any reals/complex $x,y$:
$x-2y = 3$ and $y = 4$ iff $x = 11$ and $y = 4$
This equivalence means that if your system includes the first two equations (on the left of the "iff") then you can replace them by the last two equations (on the right of the "iff") without changing the possible solutions.
If a compact notation is desired that still captures completely the "iff" nature of the transformation, I could suggest:
$x-2y = 3 \xleftarrow{}\!\xrightarrow{\displaystyle\!\!\!\! y = 4} x = 11$.