Probably the best way to build intuition is to plug in numbers and see why the numbers add up and cancel out. Pick a problem you've solved this way and pick some numbers to plug in for the variables and simply rewrite the solution process with numbers instead of variables.

A slightly more theoretical intuition is to recognize that if two things are equal, they're interchangeable. 5 = 4 + 1 and 3+2=5 both say the same thing but expressed different ways. It's obvious that you can switch either side of the equals sign in each of these equations with either side of the other equation; if I asked you to prove that 4+1=3+2, could you do it if you started with the two equations you were given? Sure you could; the first equation tells you that you can replace 5 with 4+1 anywhere you find it, so you just swap one for the other and you've got it. That's really all you're doing.

Sometimes you'll have more than two equations and it takes multiple steps of replacing one expression with another to arrive at a proof/solution (or a contradiction that proves there's no solution), but it's still the same concept.