GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 18 Mar 2019, 20:55

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If A and B are two fixed constants such that the system

Author Message
TAGS:

### Hide Tags

GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 902
If A and B are two fixed constants such that the system  [#permalink]

### Show Tags

18 Feb 2019, 14:42
00:00

Difficulty:

25% (medium)

Question Stats:

79% (02:21) correct 21% (02:19) wrong based on 19 sessions

### HideShow timer Statistics

GMATH practice exercise (Quant Class 2)

$$\left\{ \matrix{ \,2\left( {x + y} \right) - A = 0 \hfill \cr \,3x + By - 6 = 0 \hfill \cr} \right.$$

If $$A$$ and $$B$$ are two fixed constants such that the system of equations given above has more than one ordered pair $$(x,y)$$ in its solution set, what is the value of $$AB$$ ?

(A) 10
(B) 12
(C) 14
(D) 16
(E) 18

_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

Senior Manager
Joined: 04 Aug 2010
Posts: 383
Schools: Dartmouth College
Re: If A and B are two fixed constants such that the system  [#permalink]

### Show Tags

18 Feb 2019, 15:50
1
fskilnik wrote:
GMATH practice exercise (Quant Class 2)

$$\left\{ \matrix{ \,2\left( {x + y} \right) - A = 0 \hfill \cr \,3x + By - 6 = 0 \hfill \cr} \right.$$

If $$A$$ and $$B$$ are two fixed constants such that the system of equations given above has more than one ordered pair $$(x,y)$$ in its solution set, what is the value of $$AB$$ ?

(A) 10
(B) 12
(C) 14
(D) 16
(E) 18

The system will have an infinite number of solutions for (x, y) if the two equations are THE SAME:
2(x+y) - A = 0 --> 2x + 2y - A = 0 --> 6x + 6y - 3A = 0
3x + By - 6 = 0 --------------------------> 6x +2By - 12 = 0
The equations in blue will be the same if 3A=12 and 2B=6, with the result that A=4, B=3, and A*B=12.

.
_________________

GMAT and GRE Tutor
Over 1800 followers
GMATGuruNY@gmail.com
New York, NY
If you find one of my posts helpful, please take a moment to click on the "Kudos" icon.
Available for tutoring in NYC and long-distance.

GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 902
Re: If A and B are two fixed constants such that the system  [#permalink]

### Show Tags

18 Feb 2019, 18:08
fskilnik wrote:
GMATH practice exercise (Quant Class 2)

$$\left\{ \matrix{ \,2\left( {x + y} \right) - A = 0 \hfill \cr \,3x + By - 6 = 0 \hfill \cr} \right.$$

If $$A$$ and $$B$$ are two fixed constants such that the system of equations given above has more than one ordered pair $$(x,y)$$ in its solution set, what is the value of $$AB$$ ?

(A) 10
(B) 12
(C) 14
(D) 16
(E) 18

Mitch´s nice solution is based on the fact that (A and B already fixed,) the solution sets of ordered pairs (x,y) that satisfy each equation alone represent (each one) a line in the rectangular coordinate system.

We have two lines in the plane, hence they are concurrent (just one point (x,y) in common), parallel and distinct (no point (x,y) in common) or parallel and coincident (every point (x,y) in common).

From the question stem, we are looking for the last scenario, and Mitch´s solution is validated.

Let me present an alternate solution, using not only algebraic operations, but only algebraic arguments/insights. (We offer this problem in our SECOND class... no Analytic Geometry yet!)

$$? = A \cdot B\,\,\,\,\left( {{\rm{system}}\,\,{\rm{with}}\,\,{\rm{more}}\,\,{\rm{than}}\,\,{\rm{one}}\,\,{\rm{solution}}} \right)$$

$$\left\{ \matrix{ \,2\left( {x + y} \right) - A = 0 \hfill \cr \,3x + By - 6 = 0 \hfill \cr} \right.\,\,\,\,\, \cong \,\,\,\,\,\left\{ \matrix{ \,2x + 2y = A\,\,\,\left( { \cdot \,3} \right) \hfill \cr \,3x + By = 6\,\,\,\left( { \cdot \,2} \right) \hfill \cr} \right.\,\,\,\,\, \cong \,\,\,\,\,\left\{ \matrix{ \,6x + 6y = 3A\,\,\left( * \right) \hfill \cr \,6x + 2By = 12 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,y\left( {6 - 2B} \right) = 3\left( {A - 4} \right)\,\,\,\,\left( {**} \right)$$

$$6 - 2B \ne 0\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,y = {{3\left( {A - 4} \right)} \over {6 - 2B}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,x = {1 \over 6}\left\{ {3A - 6\left[ {{{3\left( {A - 4} \right)} \over {6 - 2B}}} \right]} \right\}\,\,\,\,\mathop \Rightarrow \limits^{A,B\,\,{\rm{given}}} \,\,\,\,\left( {x,y} \right)\,\,\,{\rm{unique}}\,\,{\rm{solution}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{impossible}}$$

$$6 - 2B = 0\,\,\,\, \Rightarrow \,\,\,\,B = 3\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,A = 4\,\,\,\, \Rightarrow \,\,\,\,\,? = AB = 12$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

Re: If A and B are two fixed constants such that the system   [#permalink] 18 Feb 2019, 18:08
Display posts from previous: Sort by