M06-34

Official Solution:


If \(a\), \(b\), and \(c\) are positive and \(a^2 + c^2 = 202\), what is the value of \(b - a - c\)?

(1) \(b^2+c^2=225\). Not sufficient on its own.

(2) \(a^2+b^2=265\). Not sufficient on its own.

(1)+(2) Subtract \(a^2+c^2=202\) from \(b^2+c^2=225\):

\(b^2-a^2=23\).

Now, sum this with \(a^2+b^2=265\):

\(2b^2=288\).

So, \(b^2=144\), giving \(b=12\) (since it is given that \(b\) is a positive number). Since \(b=12\), then from \(b^2-a^2=23\) we get that \(a=11\) and from \(a^2+c^2=202\) we get that \(c=9\). Sufficient.


Answer: C
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The problem with your solution is that you assume, with no ground for it, that variables represent integers only. From b^2 - a^2 = 23 you cannot say that b = 12 and a = 11. For example b could be \(\sqrt{24}\) and a could be 1.
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Hi Bunuel,

In this question, if I take only the first statement and subtract the two equations given we get:-

b^2-a^2=23.
Thus possible values positive values will be- b=12, a=11.
Upon substitution in any of the two equations we get c=9.
Thus can we not answer the question using only statement 1.

Please kindly point out the flaw in this reasoning. Is it because the question does not give us that only integer values are possible?

If \(a\), \(b\), and \(c\) are positive and \(a^2 + c^2 = 202\), what is the value of \(b - a - c\)?


(1) \(b^2 + c^2 = 225\) --> insuff: because a, b, & c are positive, may not be positive integers.

(2) \(a^2 + b^2 = 265\)--> insuff: because a, b, & c are positive, may not be positive integers.

combining all 3 equations, we get the value of \( (a^2+b^2+c^2) \), the master equation.
Subtracting each equation from this master equation, we can get the individual value of a, b, & c because they are +ve.
So we can get the value of (b - a - c)
Answer: C

In general, there's no quick way to make a decision about 3 (or more) equations in 3 (or more) unknowns. As soon as you get beyond two equations/two unknowns, things are very complicated, and the standard techniques involve matrix algebra (a subject you normally learn in the first year of an undergraduate math degree). Those techniques aren't quick, and they're way beyond the scope of the GMAT.

So in general, if you see three equations in three unknowns on the GMAT, the only way to know if you'll get one solution, or more than one, is to solve. There's no instant way to tell if one equation duplicates the information from one or both of the others, because one equation can be a combination of the other two. For example if you have these equations:

a + b = 5
a + c = 3
2a + b + c = 8

the third equation is the sum of the first two. So it's not new information, and these equations will have infinitely many solutions. Maybe that is a bit obvious, but you could instead have

a + b = 5
a + c = 3
2a + 5b - 3c = 16

and now it's not obvious at all that the third equation is a combination of the first two, but it is (if you multiply the first equation by 5, and the second by 3, and subtract, you get the third equation). Here again you'd have infinitely many solutions, but the only practical way to discover that is by solving the equations and seeing what happens. And with GMAT equations, that's going to be faster than any of the techniques you'd learn in advanced math.

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Tricky question but a good one. -- I also had assumed that B = 12 , A = 11 C = 9.

good question..I also did the same mistake of assuming the numbers as integers.

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.
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Fairly meaty question. The amount of quadratics present can be overwhelming. I got this one incorrect.

Once you get one value of any variable you can find the rest. That saves a bit of time!

If there are 3 variables and 3 equations, do we need to solve?
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IanStewart
When I see 3 equations, and 3 variables, I usually check manually that information is not duplicate in multiple equations. E.g. Equations are unique with each other.


Please share your thoughts how do you make a quick decision on seeing n equations with n variables.

Thanks!

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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