Question

Instructions: Drag each tile to the correct location on the algebraic problem. Not all tiles will be used.

Answer 1

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.)

Answer 2

4x - 7 = -2x + 12                         //GIven

4x - 7 + 2x = -2x + 12 + 2x         //Addition property of equality

6x - 7 = 12                                //Simplification

6x - 7 + 7 = 12 + 7                    //Addition property of equality

6x = 19                                     //Simplification

6x/6 = 19/6                               //Division property of equality

x =  19/6                                  //Simplification