If we went left to right and that was the only rule, then we could not write things like x raised to the (b+c) power. How would we write: x divided by the quantity (a-b)? x/a-b would mean x divide by a .....minus b so would not work. x/(a-b) does the trick, but clearly is not done left to right as you'd like....so, we have agreed upon a certain order that allows more flexibility.
Solving equations, I think, would be a nightmare as well....
2 + 3x = 14 would mean 5x=14
suddenly the commutative property would be in jeopardy as we SHOULD be able to change the above equation to:
3x + 2 = 14 ..........but now this does not mean 5x=14 anymore