Question
A car is moving with constant velocity so that its position coordinate x at time t is equal to x = at + b, where a and b are constants. Solve this equation for t. (Algebra.)
Answer
The answer is t = x/a - b/a. One can obtain this result as follows.
A basic rule in algebra is that you can do anything to an equation as long as you do the same thing to both sides of the equation (and, of course, as long as you are carrying out a "legal" algebraic operation).
Let us practise this principle step-by-step for the given equation. Since we want to end up with an equation that has t on the left-hand side and everything else on the right-hand side, we start by rewriting the equation in the form
The principle used here is that A = B implies B = A, where A and B can stand for any algebraic expressions. Now t is on the left-hand side, but it is not alone on the left-hand side. We have to get rid of the quantities a and b that appear on the left-hand side. Which one shall we gid rid of first, a or b?
It takes either a certain amount of "calculating ahead in your head" or practice to know what is going to be easiest. Here, getting rid of b on the left-hand side looks easier than getting rid of a because all we have to do to get rid of b is to subtract b from both sides of the equation. Subtracting b from the left-hand side, and using the associative law of addition, transforms the left-hand side as follows:
Subtracting b from the right-hand side of equation (1) gives x - b. Equating this to expression (2) gives
All that is left to do is to get rid of a on the left-hand side. To do this, we divide both sides of the equation by a. The result, using the distributive law of division, is
If we had tried to get rid of a first, by dividing both sides of equation (1) by a, the second step would have involved subtracting b/a and, of course, the final result would have been the same. In this case, the amount of work would have been very much the same. It would be a good exercise for you to rederive result (4) this way. Sometimes, one has no choice in which step to take first.