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Manipulating Algebraic Equations and Expressions with Fractions: Equations

A student writes,

Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better?

This is a surprisingly subtle question. It turns out that the answer is partly “Do what’s mathematically sound,” but mostly “Do what’s useful.” It’ll take me 6 posts to show what that means:

The first posts in this series concerned evaluating expressions. The rest of the series will concern manipulating equations.  This is also where we turn from “Do what’s mathematically sound,” to “Do what’s useful.”

When you’re solving an algebraic equation, what’s generally useful is to clear away all the grouping symbols, that is, the parentheses, division bars, and radical signs. Doing so allows you to move around the variables and constants in whatever way you like. There will be exceptions, but your default move when you see an algebraic equation with fractions should be to multiply every term by the least common denominator to clear the fractions, thus clearing grouping symbols.

Consider this problem:



A Nice Efficient Solution

Multiply each term by the least common denominator, 12x.


Next simplify each expression.


Once you’ve cleared the grouping symbols, you’re much more likely to be able to solve the problem. In this case, we’ll transpose, subtracting 3x from each side of the equation.


Finally , we’ll divide each side by 7 to get,



A Messy Solution that Invites Error

I’ve often seen students tackle problems like this by treating the expressions on each side of the equal sign as inviolate. That is, many students begin by adding:


This might work out alright for you in the end—there’s certainly nothing in these operations—but it’s just not very useful.


Another Solution that’s Not Quite As Messy

The same is true to a lesser extent for the method that begins by transposing to isolate the x term:


If you followed this step by immediately clearing the fractions, no harm done. Unfortunately most people who begin this way instead proceed to simplify the left hand side of the equation:


Again, this could work out alright, but a surprisingly large number of people who start this way choose answer (A) 7/12.



-If the question gives you an equation, it’s almost certainly easiest to do the same the thing to both sides of the equation, rather than to simplify the expressions on each side of the equation.

-If that equation includes grouping symbols—parentheses, division bars, radical signs—it’s usually simplest to begin by getting rid of those grouping symbols.


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