Section 1.5 Going Deeper: Clearing the Denominator
For the problems in this section, it was possible that you ended up with a fraction in your final answer, but you were not given any fractions in the original problems. Students tend to have a negative relationship with fractions. Later in the book, we're going to take a very close look at them. But for now, we're going to focus on just some basic algebra involving fractions.
Many students are at least vaguely familiar with something called "cross multiplication." This is a technique for working with the specific situation when you have two fractions equal to each other and you're trying to solve for the variable. Here is how it's often portrayed:
Many teachers teach this method, but there are some downsides to it. It creates an additional rule for students to remember, it's often applied incorrectly, and it doesn't teach or reinforce any specific algebraic concepts.
Fractions in equations introduce their own set of difficulties because fraction manipulations involve their own concepts (for example, common denominators and reducing) which make things more complicated. Unfortunately, this leads to the tendency to create even more manipulations for students to learn so that you have one set of rules for fraction equations and a different set of rules for non-fraction equations. (If you're not comfortable with fractions, there's a very brief fraction review in Subsection 15.1.1, and a more thorough discussion that runs through Section 17.1, Section 18.1, Section 19.1, and Section 20.1.)
If you look back at Definition 1.4.1, you might notice that there's one algebraic step that we did not use in this section. We never multiplied both sides of the equation by a value in our attempts to solve for the variable.
A very broad category of algebraic manipulations is known as "clearing the denominators." And this technique is built on multiplying both sides of the equation by the same value. The basic premise of this method is that it's often much easier to solve equations when there aren't fractions, and so it can make sense to eliminate them from the equation as the first step.
You can argue that cross-multiplication is "faster" in this case, and you would probably be correct. But that doesn't mean that cross-multiplication isn't without its drawbacks. The blind application of cross-multiplication can lead to numbers that are larger than necessary. Consider the following:
If we were to use cross-multiplication, we would have ended up with \(40x = 36\text{.}\) Then after dividing, we would have had to reduce the fraction, which creates more opportunities for error.
And then there's the challenge of more complex equations. Consider the following:
Students that have learned cross-multiplication will often try to apply it to this situation, even though it doesn't apply. When students learn rule-based mathematics, they will go through all sorts of interesting machinations to try to apply a rule where it doesn't belong because they simply don't know what else to do.
We could solve this using fraction manipulations, but most people (including instructors) prefer not to do that if they don't have to. And clearing the denominator is the way around that. The trick is to multiply both sides of the equation by a number that causes all the denominators to cancel out.
This technique is used much further along in mathematics. For example, there's a technique called "partial fraction decomposition" that is used in calculus to break apart a fraction into simpler components. It comes down to working with an equation that might look like the following:
The goal is to solve for \(A\) and \(B\text{.}\) A common first step is to clear the denominators so that you don't have to deal with fractions, and that manipulation looks very similar to the one above.
This is only the first step of this part of the problem. In practice, the next step would be to determine the values of \(A\) and \(B\text{,}\) then use this equation to substitute for the integrand of an integral, which then needs to be integrated.
Fortunately, you don't need to worry about this right now. The point is that the algebraic techniques you're learning right now are the same algebraic techniques you're going to see down the line, especially if you are on a track that's taking you towards calculus. It is important to do your best to build a solid foundation now so that you will be ready when you see more complicated manipulations in the future. It is an incredibly difficult task to do both at the same time.