GRE quant question walkthrough: Systems of equations (algebra)

Tyler York

In this video, Achievable GRE course author Matt Roy explains a way to quickly solve seemingly difficult algebra problems that involve two variables and two equations. He will show you how to determine if you should use the combination method and in what ways the method can be used.

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Full GRE quant question walkthrough: Systems of equations (algebra) video transcript:

This next problem is an algebra problem, and more specifically we can call it a systems of equations problem. In systems of equation problems, we have multiple variables and multiple equations. Generally they ask for the value of one of the two variables, or they ask for the value of an expression like in this problem, they actually ask for the value of X + y.
Now, whenever they ask for the value of an expression, it is very often the case that you should be using something called the combination method. That really just means combining the two equations together by either adding them to each other or subtracting one from another. In this case, you should subtract the bottom equation from the top equation, but let me explain why.
If we are looking for the expression X + y, we are looking to try to make when once we've combined them on the left side just an X&AY and on the right side whatever that difference is. In this case it would be 3. If we were to add the the bottom from the top we would get eleven X + 5 Y now.
Let me show you how this looks. You should write it out like this. We put it all in parentheses so you can see that everything should be negative because we're subtracting it Six X -, 5 Y or, sorry, six X -, 5 X is just X and three y -, 2 Y is just Y, and 16 -, 13 is 3.
We're automatically left with the expression they're looking for, and the answer would just be 3. Now, if you've never done this before, let me explain just for a moment why this is even possible. In algebra, you're allowed to do the same thing to one side of the equation as you can do to the other. Now it looks like in this problem that we're not doing that.
We're actually subtracting 5X plus 2Y from one side of the equation and subtracting 13 from the other. However, five X + 2 Y is 13. They equal each other. So even though we're subtracting something that looks different on one side as it looks on the other, because they equal each other, we are essentially subtracting from the top equation the same thing on both sides.
But because they're written differently, it allows us to manipulate the variables and that's that's the point of this problem. Whenever the question is looking for an expression, it is very often the case that you should be thinking, how can I manipulate these two equations? Which one do I need to subtract from the other? Or should I just be adding the two in order to make an expression that looks like the expression they're looking for?
And that's how you should do it.
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