7. Solve for   x   and check.

This equation is designed to illustrate that the steps for solving equations will work even when the problem looks a bit different than problems with which the student may be familiar.

  1. Clear denominators. Multiply both sides by a common denominator.
  2. Simplify. Remove parentheses and combine like terms.
  3. Transpose known terms to one side and unknonwn terms to the other.
  4. Combine.
  5. Divide both sides by the coefficient of the unknown.
  6. Check.

To clear denominators, multiply both sides by   cx + d.

This amounts to cross multiplying

ax + b = cx + d

There are no parentheses to remove or like terms to combine, so we are ready to transpose known terms to one side and unknown terms to the other. It will make no difference to which side we transpose the terms, so let's transpose known terms to the right and unknown terms to the left. If a term has a factor of   x   in it, it is an unknown term. If not it will be a known term.

ax - cx = d - b

Now comes the combine step. To combine unknown terms, factor out the common unknown factor.

x(a - c) = d - b

After transposing all of the terms that have a factor of   x   to the left, and all of the terms that do not have a factor of   x &nbdsp; to the other side, all of the terms on the left will have a factor of   x,   so we will be able to factor it out. This will get all of the   x's   together in one place with only one thing happening to it. It is being multiplied by   (a - c).   To get rid of multiplying by   (a - c),   divide by   (a - c)

When we get a solution, we need to check for zero denominators. The denominator will be zero if   a = c.   If   a = c,   then the equation would become

or

ax + b = ax + d

and after subtracting   ax   from both sides, we are left with

b = d

also. In that case the solution would be of the form  0/0.   0/0   is different from something nonzero over zero.   6/0   would be how long it would take to travel 6 km if you don't move. It would take forever to travel 6 km if you don't move. That would give you and answer of infinity. Something nonzero over zero is infinity.

0/0   is a different story. That would be how long it would take you to stay where you are if you don't move. If you don't move, you will be where you started after 1 hour, after 2 hours, after 3 hours, etc. This illustrates that   0/0   could be anything.

In this case, the fraction in the equation would have the same thing on both the top and bottom and would be equal   1   no matter what   x   was. This is consistent with the fact that   0/0   could be anything.

Now we get to check the result. Copy down the original equation except instead of copying down an   x,   copy down the solution in parentheses.

This gives us an exquisite compound fraction on the left. Find common denominators for the top and bottom of the fraction.

and it does check.

Notice, in what we wind up with

there are two factors in the bottom. One is  a - c,   and we have discussed what happens if this is zero. But there is also another factor,   ad - bc.  This comes up in a surprising nuumber of cases. It will be zero if

ad = bc

Divide both sides by   bd.

or

which is to say that if   ad = bc   then the ratio of   a:b   is the same as the ratio of   c:d   unless b or d are zero. To see the effect that this has, note

If

let

Then our equation becomes

If we cancel the   rx + 1   from top and bottom, we get

or

b = d

If   b = d   and ad = bc,   then we also have   a = c,   which we have already discussed. To make a long story short, if   ad = bc   then one of two things wiol hapen. If, also,   b = d,   then we have an identity, and any value of   x   will be a soution. If   ad= bc   but   b   is not equal to   d,   then there will be no solution.

When solving equations with literal coefficients, we always to consider these possibilities.