I still remember the horror I felt at my first exposure to simultaneous linear equations. You know, this sort of thing:
2x + y = 7
3x - y = 8
I couldn't believe my ears when the teacher suggested we should just add the two equations together. How could this be? I thought it would violate the laws of God and man alike. It seemed completely arbitrary and inexcusable. It was a real "turn the giraffe upside down" moment for me.
Over time, I came to realize that it's OK to add the equations together because each equation represents two equal values. If 2x + y = 7, you can add the same value to both sides of the equation and the equals sign will still be true. In this case, the value we will add to both sides of the equation is contained by the second equation, 3x - y = 8. The equals sign means that 3x - y is the same value as 8, so we can 3x - y to one side of the first equation and 8 to the other side of the first equation, and the equals sign will still be true.
The power of the equals sign is a tricky concept. I recently showed my older daughter how to simplify one side of an equation, and she asked, "does this mean I have to do something to the other side?" I said no, because I hadn't changed the value of the expression that I simplified, so the equals sign still held true. I can tell she will need more work on this point.
This is where curricula like Trailblazers make a fundamental mistake. For some obscure reason, they explicitly discourage writing mathematical notation, in favor of either "mental math" or writing paragraphs explaining how you reached the solution. But it is essential that students practice writing the symbolic language of mathematics, which was developed over centuries with the exact purpose of expressing mathematical ideas in the most succinct, clear, and concise way possible.