The following two details should be quite easy to remember.
1) "Do the same on both sides." (Whatever you do on one side of the equation, you must do the same on the other side.)
2) "Isolate the variable." Though you can do just about any operation, you should do operations in such a way that the variable which you are solving for will be alone on one side; anything else, on the other side.
All else is practice, and learning some special cases.
Tell me the equations first.
You need as many equations as you have variables.
I have taken Algebra I and made an 99 easily. You must learn to memorize how to solve equations and various formulas. Be prepared to right out long equations and use substitution and elimination to simplify equations. Also, pay attention in class.
You can solve the system of equations with three variables using the substitute method, or using matrix operations.
If you don't learn to solve equations then guess and check is the only way to arrive at new information.