5. Solve for   x   and check

This is a rational equation, so we run through the steps for solving rational equations. In this case the coefficients are letters, so when we run through the steps, we will be doing arithmetic with letters. But that's all right, because we can do arithmetic with letters. It's called "algebra."

First clear denominators. There is only one denominator, namely   x - 1,   so multiply both sides by   x - 1.

or

a = b(x - 1)

Notice that when we move a factor out of one side, if it started out on the bottom it ended up on the top of the other side. Remove parentheses.

a = bx - b

At this point we transpose all unknown terms to one side and known terms to the other. While all of our terms have letters, since we are solving for   x,   a term that has a factor of   x   will be considered an unknown term, and if a term does not have a factor of this unknown, we will consider it to be a know term. Transpose the   b   to the left.

a + b = bx

At this point we have only one   x,   and the only thing that is happening to it is it is being multiplied by   b.   Divide both sides by   b.

or

Notice that when we move a factor from one side to the other, if it starts out on the top it ends up on the bottom when it moves to the other side of the equation.

Check. Copy down the original equation

except that wherever you see an   x,   copy down the solution in parentheses. In this case, since the equation has some letters for coefficients, the check will be a bit more extensive. However, the set up step is very straightforward. Copy down the original equation except that wherever you see an   x,   copy down the solution in parentheses.

This gives us a compound fraction on the left. Subtract the fractions on the bottom. This will require a common denominator. The common denominator will be   b.

Now that we have common denominators, we can subtract the tops in the bottom.

which simplifies to

We are ready to invert and multiply.

giving us

b = b

and it checks.

The checks to these rational equations with literal coefficients provide us with excellent opportunities to practice our principles for simplifying rational expressions. Since we have solved to find the expression which we can substitute for x and have the equation come out right, we get problems which are cooked up so that amazing things happen.

The student shouldn't get the idea that the steps we have outlined are the only way to do the problem. The reason that we present the steps is because they will work best most often. That does not mean that there are not other ways to solve these equations or that there might not be a better way in some instances. In this case there is only one occurrence of the unknown. Whenever that happens, all you have to do is to undo all the arithmetic that is happening to the unknown. If we look at the original equation

one of the problems here is that the unknown is in the bottom. We could fix that by taking reciprocals of both sides.

Now simply multiply both sides by   a.

Now we just need to add 1 to both sides.

This looks a little different from the solution we got before, but is really is the same thing. Since we are adding fractions, we could find common denominators,

we get the same answer we got before.

Notice that both answers check. If we substitute the other form of the answer in the original equation

we would get

which would actually check more easily. Subtract in the bottom on the left side,

and we are quickly ready to invert and multiply.

which checks.

b = b