Algebra Mini-Series #1: Replacing Large Expressions with a Single Variable

This is part one 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!

Five Tricks to Improve Your Algebra Skills

Part 1: Replace Large Expressions with a Single Variable When Possible

Sometimes you might have to solve a particularly nasty looking equation which doesn’t actually require much difficult algebra, but is nonetheless difficult due to its complexity. This is true in practically all high school and college math, as well as in chemistry, physics, and other disciplines. If you’re going to be copying down ugly numbers again and again, why not replace them with a simpler placeholder until you aren’t ready to calculate everything together at the end? Consider the following example:

Solve for y:
(x+√(1/x+x²))*y + (x³+2)/y = y

If we replace everything besides our y variables with the following, the equation takes a much nicer form.

Let:
a = (x+√(1/x+x²))
b = (x³+2)

Then we just need to solve:

ay+b/y=y
ay-y=-b/y
(a-1)y = -b/y
(a-1)y² = -b
y² = -b/(a-1)
y = ±√(-b/(a-1))
y = ±√(-(x³+2)/(x+√(1/x+x²)-1))

Of course, this answer isn’t nice at all, though if we hadn’t swapped out its components for simpler variables, it would have been even harder to solve by hand without making a mistake. As an added bonus, we don’t have to copy down a and b each time we write down a step, which saves time—and on standardized tests such as the SAT, or any other math test as well, time is invaluable.

Of course, the one down side of this technique is that it might be confusing at first to have so many variables in your equation, but this isn’t really an issue for a variety of reasons:

• As math classes get more advanced, this happens anyway, so if you’re not already used to this, it’s best to prepare early, which will give you a leg up on your peers.
• Relatively speaking, single letters are much nicer than the nasty expressions they replace.
• They’re a lot quicker to write, and it’s identical to algebra with normal variables.

Just remember the basic technique—replace coefficients and expressions which do not include the variable of interest (in the above case, y, but often x) with letter variables, solve, and then back substitute. If you’ve taken integral calculus before, you’ll recognize this as a simpler but just as powerful version of u-du substitution.