Just as we can add and subtract constants from both sides of an equation, we can also add and subtract copies of the variable from both sides of the equation. Therefore, if the same variable appears on both sides of the equation, we can reduce them as much as possible in order to get one variable on one side of the equation. It's always nice to have just a single "x" (especially when following a treasure map, you know, as you do). We need to add or subtract the same number of copies of the variable from each side.
Remember that our mission, if we choose to accept it, is to get the variable on one side of the = sign and a number on the other side.
Solve the equation 4x = 5x + 1. Check your answer.
We'd like to have all the x's by themselves on one side of the equation, so we subtract 4 copies of x from each side to find that
0 = x + 1
Yay—so few copies! This will shave a bundle off our Kinko's bill.
We know what to do from here: subtract 1 from each side of the equation, and write -1 = x.
To check our answer, we evaluate the left side of the original equation and the right side of the original equation individually for x = -1. The left side of the equation, evaluated at x = -1, is
4(-1) = -4
The right side of the equation evaluated at x = -1 is
5(-1) + 1 = -5 + 1 = -4
Because the two sides of the equation agree when evaluated at x = -1, the solution to the equation is indeed x = -1. There's one of those negative solutions again. Sorry, Diophantus.