The steps involving properties of equality are shown in painful detail.
For Step 4, you add 7 to both sides of the equation (with the goal of cancelling the -7 on the left). So you have
6x -7 +7 = 12 +7 . . . . . . . showing 7 added to both sides
In the next step, this is simplified to the way you might have written it to start with:
6x = 19
There should be no confusion in any of this. All solutions of all equations make use of the properties of equality like this. Addition is used to cancel subtraction, division is used to cancel multiplication. (Root-taking cancels powers, and vice-versa; inverse trig functions cancel trig functions; and so on.)
For Step 4, you add 7 to both sides of the equation (with the goal of cancelling the -7 on the left). So you have
6x -7 +7 = 12 +7 . . . . . . . showing 7 added to both sides
In the next step, this is simplified to the way you might have written it to start with:
6x = 19
There should be no confusion in any of this. All solutions of all equations make use of the properties of equality like this. Addition is used to cancel subtraction, division is used to cancel multiplication. (Root-taking cancels powers, and vice-versa; inverse trig functions cancel trig functions; and so on.)