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# If P and Q are non-zero integers, is the equation Py + Qx = 7, exactly

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Joined: 21 Aug 2013
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If P and Q are non-zero integers, is the equation Py + Qx = 7, exactly  [#permalink]

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04 Jul 2018, 22:34
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55% (hard)

Question Stats:

45% (02:08) correct 55% (01:22) wrong based on 14 sessions

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If P and Q are non-zero integers, is the equation Py + Qx = 7, exactly the same equation as 9x - 6y = 21 ?

(1) P + Q = 1

(2) P*Q = -6
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Joined: 07 Dec 2017
Posts: 841
Re: If P and Q are non-zero integers, is the equation Py + Qx = 7, exactly  [#permalink]

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04 Jul 2018, 22:44
amanvermagmat wrote:
If P and Q are non-zero integers, is the equation Py + Qx = 7, exactly the same equation as 9x - 6y = 21 ?

(1) P + Q = 1

(2) P*Q = -6

Two equations are equal if and only if all of their coefficients are equal.
We'll look for statements that give us this information, a Logical approach.

Rewriting our equations so they both equal 7 gives:
Py + Qx = 7
3x - 2y = 7

So our equations are equal when P = -2 and Q = 3.

(1) One equation, two variables --> infinite solutions.
Insufficient.

(2) One equation, two variables --> infinite solutions.
Insufficient

Combined:
Two equations, two variables usually gives one unique solution but in this case both of our equations are symmetrical.
So, while we can find the right numbers (they happen to be 3 and -2), we don't know which is P and which is Q.
Insufficient

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If P and Q are non-zero integers, is the equation Py + Qx = 7, exactly  [#permalink]

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05 Jul 2018, 04:02
amanvermagmat wrote:
If P and Q are non-zero integers, is the equation Py + Qx = 7, exactly the same equation as 9x - 6y = 21 ?

(1) P + Q = 1

(2) P*Q = -6

Given, Py + Qx = 7----------(a)
& 9x - 6y = 21 Or, 3x + -(2)y =7 ---(b)

Equating x & y co-efficient in (a) and (b) we have P=-2 & Q=3

So, when P=-2 and Q=3, both the equations become same.

Re-phrased Question stem:- Is P=-2 ,Q=3 ?

Statement-2
P + Q = 1
When P=-2 & Q=3, our answer to question stem is Yes, but there are 'n' number of combinations for which P+Q=1.
Hence, insufficient.

Statement-2
P*Q=-6
When P=-2 & Q=3, our answer to question stem is Yes, but there are 'n' number of combinations for which P*Q=-6.
Hence, insufficient.

(1)+(2), we have

$$(P-Q)^2=(P+Q)^2-4*PQ$$

we know the value of P+Q & PQ from st1 & st2 respectively, therefore, we can calculate P-Q & subsequently, value of P & Q can be determined.

So, ($$P-Q)^2=1^2-4*(-6)=1+24=25$$

So, P-Q=5 Or, P-Q=-5

P-Q=5 & P+Q=1. yields P=3, Q=-2...In this case answer to question is No.

P-Q=-5 & P+Q=1. yields P=-2, Q=3...In this case answer to question is Yes.

So, insufficient.

Ans (E)
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If P and Q are non-zero integers, is the equation Py + Qx = 7, exactly &nbs [#permalink] 05 Jul 2018, 04:02
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