How many equations to find a variable?

Question

I remember from school that we need as many equations as unknown variables to be able to find the values of unknowns.

I just came across a DS problem where they were able to find a value of a variable using two equations in a three variable problem.

So now I'm wondering, I know we need 3 equations to solve for 3 variables. Are there any rules of how many equations we need to find 1 or 2 out of the 3 variables?

AndrewN · Answer

Hello,  newdimension , and welcome to the forum. There are times in which you can look at two multivariable equations with three unknowns to solve a question asking about one of them, but a certain framework would have to be in place to allow you to answer the question. An example from a DS question I am making up on the spot:

What is the value of z?

(1) x + y + z = 22

(2) 3x = 42 - 3y

In this case, you could definitively answer the question--z = 8--whether you had substituted algebraically into the first equation or had manipulated the second to eventually read, x + y = 14 and then substituted out x + y in the first equation. You will never know what either x or y may be individually, but you can pin down how they interact when added, and that is all that matters to answer the question being asked.

I would advise you to focus on getting the fundamentals down before worrying too much about shortcuts. Always ask yourself the following:

1) What is the question asking?

2) What information do I have?

3) What is the core concept being tested?

Together, this three-step process can help you get the ball rolling on just about any problem, once your foundational knowledge is in place, or can help you understand that you can get the answer, in some cases, without doing all the work. You might call that a shortcut, but I prefer to think of it more as a way to use the test against itself. The GMAT™ tests analytical  reasoning  ability, not necessarily mathematical prowess. (You can read all about the theory by following the link in the signature of any post by   Bunuel   to the Ultimate GMAT Quant Megathread.)

Good luck with your studies.

- Andrew

shameekv1989 · Answer

There are some constraints as well that can determine the values of variables without having a need to have multiple values.

For eg - Diophantine equations 
This is when you need to find Integral solutions and we have constraints on these variables about the maximum and minimum values, these can take.

For eg. 2x + 3y = 5 
The only positive integer solutions that x and y can take are x=1; y=1

Hope it helps!

newdimension · Answer

Hi Andrew,

Thank you for the feedback. The reason I asked this question is because I noticed that whenever I came across a 3 variable problem with 2 equations I would automatically think we don't know enough to solve (NS). I just noticed that isn't true, because sometimes we just want 1 of the variables.

AndrewN · Answer

Yes, that is true. The GMATTM is a test that is good at trapping unwitting test-takers in their own assumptions, whether the issue is an assumed  integer , an assumed  positive  integer, or, as was the case above, an assumed impossibility. Read carefully and do not be afraid to slow down at first while you work on building your foundational knowledge. The more you practice, and the more you read about how others have approached similar questions, the better you will get at developing your own techniques to tackling whatever problems the test may throw at you.

If you come up with other questions at any point along the way, feel free to post. The community would like to see you succeed.

- Andrew

How many equations to find a variable?

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