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:
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Isn’t 15 (b.) because:
(p^2)+(r^2)=18 —> (p-r)(p+r)=18—>(2)(p+r) = 18 –> p+r=9
You’re absolutely right, and I’ve changed it. That’s an embarrassing vestige from a previous version of this post and I hang my head in shame. I owe you a great debt. 🙂
How would you go about solving #19(the system of equations)?
It’s a little different from the examples.
Thanks for asking!
How do you get from what they’ve given you to what they want? You’re going to have to get rid of the y’s and z’s, since they only asked for x. If you add all the equations together, the y’s and z’s cancel out, and you get 7x = 140. So x = 20!
This article – your site in general – blew my mind on SAT prep , you are greatness!
thank you this really helped
🙂 I aim to please.
How about this one from the diag?
If x2 + y2 = 14 and xy = 3, then (x + y)2 =
Well, (x + y)^2 = x^2 + 2xy + y^2.
x^2 + y^2 + 2xy
14 + 2(3) = 20
Help on #8 for diagnostic test 3?
Add those bad boys up! You get 5y + 5x = 30. Since you want y + x, divide both sides by 5 and you get y + x = 6.
Could you please explain 15?
The key is recognizing that p^2 – r^2 is the difference of two squares, so it can be factored to (p + r)(p – r). So basically, you’ve got (p + r)(p – r) = 18, and p – r = 2. So substitute to solve:
(p + r)(p – r) = 18
(p + r)(2) = 18
p + r = 9
Hi Mike, I have a question about this one, too. If you don’t recognize that it’s the difference between two squares and try, instead, to eliminate one variable, it still should work out, right? If p-r = 2, then p = r+2. So we should be able to substitute (r+2) for p. After FOIL, etc, the equation comes to 22, not 18. Did I miss something?
That should still work. Here’s my algebra:
Thanks, Mike! Simple arithmetic error (r= 18/4, which led to awkward squares) blinded me.
I ran into a problem w/ #19. Off the bat, I made the mistake of not realizing that the problem was so simple as adding the equations, so I opted to subtract. I expected, however, that this would lead me to the same answer. I did this:
[4x + y – 2z = 12]
-[2x – y + z = 36]
-[x + 2y – 3z = 92]
and got x = -116 (via 12 – 36 – 92). So, either I screwed up some simple algebra/arithmetic or subtraction in the first place inevitably led me to the wrong answer. Could you elucidate?
Phew…you gave me a heart attack here for a minute. But you flipped your signs in the first equation! The first one should read 4x – y + 2z, not 4x + y – 2z.
If you put my 3 equations into Wolfram Alpha for a quick solve, you’ll see that the only solution is x = 20. You’ll also see, if you put in the 3 equations you typed, that your solution works. Because they’re different sets of equations. 🙂
I have a huge problem with solving for expressions. I can’t seem to figure out how to start off the question.
It takes a lot of practice, but getting started usually requires adding, subtracting, multiplying, dividing, or substituting. I know that’s a lot of things, but if you think about all of them when you begin a problem, the first step will start to emerge for you.
Does it matter if you add or subtract the equations in #19?
Sure does! If you add them all up, the y and z terms cancel out and you’re left with 7x = 140, which of course simplifies to x = 20.
The key to questions like these is to look at them for a minute before you start adding or subtracting. What do you need to do to the system to get rid of the things you don’t want?
whats the difference between p^2-r^2 and (p-r)^2?
A HUGE difference! If you FOIL out (p – r)^2, you’ll see.
See the attached images for additional clarification.
oh okay! I thought so I think its because i look too quickly to realize how different they are. i guess this is a trick the SAT uses? Thank you!
I don’t know if I’d call it a trick, but both definitely appear on the SAT, so you should always read them very carefully and make sure you’re working with the right one. 🙂