Algebra Mini-Series #2: Moving Quantities Left and Right in Equations

This is part two of our weeklong series on improving your algebra skills. Our goal for this series is to provide a useful resource for both students and teachers, so that this article can be used in the classroom, for test prep, or to help yourself practice and master skills you never learned. With these tips, we hope you can make your algebra cleaner, faster, and more intuitive. Check back tomorrow for another informative article on some aspect of elementary algebra!

Back to: Part 1: Replacing Large Expressions With a Single Variable

Forward to: Part 3: Using Substitution to Solve Equations

Five Tricks to Improve Your Algebra Skills

Part 2: Move Quantities Left and Right Instead of Adding to Both Sides

What you’re about to learn is one of the most fundamental ways many effective students differentiate themselves from average students. This technique might take some getting used to, but it is so important that it can mean the difference between being enjoying learning math and having to put in many frustrated hours just to keep up with your peers. So what is it? Let’s begin by thinking about the following equation:

x + 5 = 7

How would you solve this? Most students initially learn to solve this equation by “adding” -5 to both sides to cancel the five on the left and preserve overall equality. Of course, this is mathematically sound, but just as though you have to take off your training wheels before you can learn to bike up a mountain, it’s likewise necessary to move on from this technique before attempting any mathematical Tour de France.

The unfortunate truth is that solving equations this way is an unnatural contrivance, a convention taught to make elementary teachers’ lives easier while eventually causing headaches for secondary teachers who wonder why their students can’t handle anything beyond linear single-variable algebra. If you ask a bright child who has not yet had the experience of learning classroom algebra the equivalent question “five plus what number is seven,” they will almost certainly think for a minute and then give you one of the following two responses:

1. 2, because 2 + 5 is 7
2. 2, because that’s 7 – 5.

Of course, the first answer simply demonstrates guessing, which isn’t going to help a student to find the roots of a cubic polynomial with irrational coefficients, but the second one, on the other hand, provides insight into how most people’s brains naturally work. The child will most likely not be able to tell you how they arrived at the second answer, but rather that that’s just the way it works. Of course, this is equivalent to subtracting 5 from both sides, but it skips this step altogether, and as a result saves time and effort.

What that child intuitively demonstrates is this: when a quantity is added or subtracted to one side of the equation, it’s easiest to isolate the variable by moving quantities from one side to the other and flipping their sign. In a sense, all we’re doing is putting the things we don’t care about in one box and leaving everything we do care about in another. While this is inherent in the “adding a number to both sides” method as well, the means does not make intuitive sense to many students since we’re adding and subtracting things which “aren’t there” with no natural end in mind. Solving our equation using our new method proceeds as such:

x + 5 = 7
x = 7 – 5
x = 2

The end result that this method is both more natural and generally more efficient. The key is twofold. First, for this problem, it doesn’t seem much faster (if at all) to use this method, but in fact, with practice, the intermediary step is completely eliminated, whereas when adding to both sides, students will never get away from writing “-5” beneath both sides of the equation, costing time and causing clutter. Second, for more complicated problems this becomes significantly faster, both for mental math and for reducing the amount of work needed on paper to solve a problem. Consider the following problem (which has only one degree of additional difficulty over the above equation) and the difference in solving it using both methods.

Old Method:
2x + 5 = x + 7
-x    -x
x + 5 = 7
-5   -5
x = 2

New Method:
2x + 5 = x + 7
x + 5 = 7
x = 2

It might not seem like much, but we’ve ultimately reduced the amount of work necessary by a factor of about one half. The more notable expression of this comes from personal experience: I have never had a student ask me for help with basic algebra who does NOT use the “old” method, and conversely I almost never have a student who DOES use this “moving left and right” method need help with algebra. If you’re trying to practice this technique for yourself, the best thing you can do is sit down with a list of simple algebra problems and run through them. With time, it will become almost automatic and will greatly increase your equation solving efficiency.

Bonus: The same is also true of multiplication and division, except instead of switching positive negative with switch numerators and denominators. However, you need to be a bit more careful. Take the following for example:

Solve for y:
3x = y/5 – 1

Clearly, y = 15x + 5, but if we’re going to move a fraction from one side to another, it needs to be the only fraction on one side, so first we have to move the -1 to the left side:

3x + 1 = y/5
15x + 5 = y