Similarly, there are two ways to solve an equation between two fractions
(a *rational equation*,
assuming that the top and bottom of both sides are polynomials,
so that each side is a rational expression).
The more direct way is to *cross-multiply*:
simplify each side separately,
then multiply the top of each side by the bottom of the other side,
setting the two resulting polynomials equal.
In other words, to solve *A*/*B* = *C*/*D*,
you solve *A**D* = *B**C*,
or equivalently solve *A**D* − *B**C* = 0.
(If you are unable to factor this polynomial,
then try factoring the original tops and bottoms;
anything that shows up in both *A* or *D* and in *B* or *C*
can then be factored out.)

Then there is another method of solving rational equations, which corresponds to the second method of simplifying complex fractions. Now you just factor the denominators, then multiply both sides of the equation by a common denominator. This is especially helpful when the equation has addition or subtraction of rational expressions; instead of simplifying each side (which also requires finding common denominators) you can multiply everything by a common denominator and make all of the fractions go away. (You still have to solve the resulting polynomial equation, of course.)

Either way, you must check for *extraneous* solutions!
Since every method of solving rational equations
involves multiplying both sides by something that might be zero,
you need to check that none of these expressions that you multiplied by
actually evaluates to zero for any of your solutions.
Equivalently, you can check that the original expressions in the equation
are both defined for all of your solutions, with no division by zero.
If any of your solutions fails this check, then you must throw it out.
(Sometimes you'll throw out all of them, sometimes none of them;
you never know until you check.)

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2019 September 3. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1100/2019FA/rateq/`

.