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Mathematical Journeys: Solve Only for What You Need

Standardized test math doesn't behave like normal math. On a normal math test, your knowledge of the concepts and material is being tested, using (hopefully) fair test questions. On a standardized test, though, they're looking for you to think outside the box, to apply math concepts and algorithms to unusual situations, and to really understand what they're looking for and find the quickest way to go about it. Let's take a question from a recent GRE student's lesson:

If 4x – 5y = 10 and 6y – 3x = 22, then what is x + y?

Now, this is a set of two equations with two variables each, so it looks to me like a perfect candidate for solving as a system. If I were solving this one on a regular math test, I'd start off trying the substitution method, since I'm more comfortable with that one. So let's explore that one first:

I'll start by solving the first equation for y:

4x – 5y = 10
- 5y = 10 – 4x
y = (-10/5) – (4/-5)x
y = -2 + (4/5)x

Then I'll plug that in for y in the second equation:

6(-2 + [4/5]x) – 3x = 22
-12 + (24/5)x – 3x = 22                    Now we have to convert the 3x into a fraction
-12 + (24/5)x – (15/5)x = 22
-12 + (9/5)x = 22
(9/5)x = 34
x = 34 (5/9)
x = 170/9

Then plug that back in for x in the first equation:

y = -2 + (4/5)(170/9)
y = 136/9

And, FINALLY, find the quantity asked for in the problem by adding x and y together:

x + y = (170/9) + (136/9)
x + y = 306/9
x + y = 34

Well, that's one way to find the answer, but that took a long time, with lots of large numbers, and lots of potential for mistakes. This is a standardized test, remember, so time is a factor here. Take a look at the question again. It's asking for x + y. Why wouldn't it be asking simply for x, or y, or even x and y, for that matter? Is it because x + y is a much cleaner number? Is it to be ornery? To make you waste time?

Well, to be honest, the answer to that last question is yes, but not in the way you might think. In our math classes, we're hardwired to try to solve for x – we want to end up with a nice clean number to equal one of our variables. It's the way most math classes work; manipulate the equation until it tells you the missing piece of information. The test builders know that, and they know that everyone's first instinct in a math problem is to try to solve for x. But in this case, they're not asking for the value of x; they're asking for the value of an expression containing x. And they're doing that very deliberately – because finding x + y is much easier than finding x.

Take a look at our system again – this time I'll re-arrange it slightly in preparation for using the addition method to solve it:

   4x – 5y = 10
– 3x + 6y = 22

See it yet? Use the addition method – don't even modify anything – and add straight down the columns:

   4x – 5y = 10
– 3x + 6y = 22
    x + y = 32

Well, would you look at that? That's the answer they're looking for – and you'll notice it's not the same answer as our previous attempt. Not only would you have wasted a bunch of time going through all those hoops to solve with substitution, but you would have gotten the question wrong to boot!

It's an odd way of looking at a math problem, but one of the biggest strategies I tell my students is to not think about the test as a math test. It's a logic test that happens to involve numbers. Here, the test is remembering to only solve for what you need. Don't bother getting all the way down to x if x won't help you in the end. Sometimes they're asking for a quantity because going any further past that quantity will only cause you grief. Solve for the quantity they ask for, and no more.
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