If I asked told you it was my birthday and I wanted a cake, what would you do? You’ve got two choices: buy a bunch of ingredients and start baking, or go to a different aisle in the grocery store and just buy the cake.
Baking the cake yourself is not only more time consuming than just buying one; it also gives you more opportunities to screw up (like, say, mistake salt for sugar and bake the grossest cake of all time). Since you know I’m a shameless crybaby who will never let you forget it if you ruin my birthday, you should just buy the cake in the cake aisle, and then use your time to do something more fun than baking.
And so it is with the SAT. What do I mean? WHAT DO I MEAN??? Read on, young squire.
Here’s a pretty common question type on the SAT:
- If 3x – y = 17 and 2x – 2y = 6, what is the value of x + y?
The SAT is asking you for a cake here. Baking it yourself will still result in a cake, but it will also give you opportunity to screw up, and take longer than just buying one. They don’t give a rat-turd if you buy the ingredients (x and y), so don’t waste time on them! All that matters is the finished cake, (the value of the expression x + y), and we should be able to solve for that directly without ever finding x or y individually.
To do so, first stack up the equations we’re given and the expression we want:
3x – y = 17
2x – 2y = 6
x + y = ?
Do you see it yet? How about now:
3x – y = 17
–[2x – 2y = 6]
x + y = ?
That’s right. All we need to do is subtract one equation from the other (I’ve already distributed the negative here):
3x – y = 17
-2x + 2y = -6
x + y = 11
Our answer is (C) 11.
Note that we didn’t have to do any algebra here (aside from distributing that negative). This is not the exception on the SAT, this is the rule. When you’re given two equations and asked to solve for an expression, you almost NEVER have to do algebra. So instead of jumping into an algebraic quagmire as soon as you see questions like this, ask yourself “How do I quickly go from what they gave me to what they want?” Let’s look at one more example together:
- If (x – y)2 = 25 and xy = 10, then what is x2 + y2?(Let’s call this one a grid-in, so no multiple choice.)
Here, we aren’t going to get anywhere by simply adding or subtracting our two equations, but do you see anything else going on? Do you see that you’ve been provided with all the pieces of a particular puzzle?Let’s start by expanding what we were given (FOIL works here of course, but you should really make an effort to memorize the basic binomial squares and difference of two squares to save you time on the test; they all show up frequently):
(x – y)2 = 25
x2 – 2xy + y2 = 25
And would you look at that—we’re pretty much already done. Just substitute the value you were given for xy, and do a little subtraction:
x2 – 2(10) + y2 = 25
x2 – 20 + y2 = 25
x2 + y2 = 45
Note again that we didn’t need to solve for x or y individually to find this solution; all we needed to do was move some puzzle pieces around. Note also that actually solving for the individual variables would have been a huge pain.
Again, your mantra: “How do I quickly go from what they gave me to what they want?“
Now YOU try: