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

 It is currently 22 Jan 2019, 06:55

### 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

## Events & Promotions

###### Events & Promotions in January
PrevNext
SuMoTuWeThFrSa
303112345
6789101112
13141516171819
20212223242526
272829303112
Open Detailed Calendar
• ### The winners of the GMAT game show

January 22, 2019

January 22, 2019

10:00 PM PST

11:00 PM PST

In case you didn’t notice, we recently held the 1st ever GMAT game show and it was awesome! See who won a full GMAT course, and register to the next one.
• ### Key Strategies to Master GMAT SC

January 26, 2019

January 26, 2019

07:00 AM PST

09:00 AM PST

Attend this webinar to learn how to leverage Meaning and Logic to solve the most challenging Sentence Correction Questions.

# Two variables - two equations trap

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 08 Dec 2015
Posts: 292
GMAT 1: 600 Q44 V27
Two variables - two equations trap  [#permalink]

### Show Tags

24 Jan 2017, 11:12
Hi everyone!

I get confused when it comes to applying the "we need two different equations to solve for a variable" rule.

Can you please give an example when just having one eq with two variables is enough to solve?

Thank you!
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4485
Re: Two variables - two equations trap  [#permalink]

### Show Tags

24 Jan 2017, 15:35
3
1
iliavko wrote:
Hi everyone!

I get confused when it comes to applying the "we need two different equations to solve for a variable" rule.

Can you please give an example when just having one eq with two variables is enough to solve?

Thank you!

Dear iliavko,

I'm happy to respond.

I will distinguish three cases:
Case I: one equation enough to solve
Case II: two equations needed to solve
Case III: two equations not enough to solve for two variables
I point out that Case II is statistically the most common case, and most often tested. That's the rule, but there are exceptions.

Case I only happens when the question is asking not for the value of x or the value of y, but simply for the value of some expression. Here's a very simple example. Suppose the question is "What is the value of y - x?" Suppose we are given the single equation 2y - x = 28 + x. We could rearrange that equation to solve for the desired expression, which equals 14, despite the fact that we cannot find the individual values of the variables.

Case II happens when we have two different equations that represent two lines with different slope. Any two lines in the x-y plane that have different slopes will intersect at exactly one and only one (x, y) point. That determines a unique and unambiguous solution.

Case III happens when the two lines have the same slope. For example:
Equation #1: x - 2y = 5
Equation #2: 4y + 6 = 2x
It's not necessarily obvious at first glance, but if you were to re-write these both in y = mx + b form, you would see that they have the same slope. If you try to solve them, you will get mathematical nonsense. In fact, they are parallel lines, and they never intersect, so there's no solution.
A variant on this case:
Equation #1: 3x + 2y = 5
Equation #2: 4x = 10 - 6y
This what mathematicians call a "degenerate system," because the two equations are really one equation in two different forms. Here, we can't solve, because absolutely every value that works in equation #1 will work in equation #2 as well, because they're the same equation! This has a continuous infinity of possible solutions, which in practice, is equivalent to having no solution.

Does all this make sense?
Mike
_________________

Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Senior Manager
Joined: 08 Dec 2015
Posts: 292
GMAT 1: 600 Q44 V27
Re: Two variables - two equations trap  [#permalink]

### Show Tags

01 Feb 2017, 02:37
Fantastic explanation! Thank you so much

Cheers!
Manager
Joined: 17 May 2015
Posts: 249
Re: Two variables - two equations trap  [#permalink]

### Show Tags

01 Feb 2017, 04:09
mikemcgarry wrote:
iliavko wrote:
Hi everyone!

I get confused when it comes to applying the "we need two different equations to solve for a variable" rule.

Can you please give an example when just having one eq with two variables is enough to solve?

Thank you!

Dear iliavko,

I'm happy to respond.

I will distinguish three cases:
Case I: one equation enough to solve
Case II: two equations needed to solve
Case III: two equations not enough to solve for two variables
I point out that Case II is statistically the most common case, and most often tested. That's the rule, but there are exceptions.

Case I only happens when the question is asking not for the value of x or the value of y, but simply for the value of some expression. Here's a very simple example. Suppose the question is "What is the value of y - x?" Suppose we are given the single equation 2y - x = 28 + x. We could rearrange that equation to solve for the desired expression, which equals 14, despite the fact that we cannot find the individual values of the variables.

Case II happens when we have two different equations that represent two lines with different slope. Any two lines in the x-y plane that have different slopes will intersect at exactly one and only one (x, y) point. That determines a unique and unambiguous solution.

Case III happens when the two lines have the same slope. For example:
Equation #1: x - 2y = 5
Equation #2: 4y + 6 = 2x
It's not necessarily obvious at first glance, but if you were to re-write these both in y = mx + b form, you would see that they have the same slope. If you try to solve them, you will get mathematical nonsense. In fact, they are parallel lines, and they never intersect, so there's no solution.
A variant on this case:
Equation #1: 3x + 2y = 5
Equation #2: 4x = 10 - 6y
This what mathematicians call a "degenerate system," because the two equations are really one equation in two different forms. Here, we can't solve, because absolutely every value that works in equation #1 will work in equation #2 as well, because they're the same equation! This has a continuous infinity of possible solutions, which in practice, is equivalent to having no solution.

Does all this make sense?
Mike

Hi,

Thank you very much, Mike, for such a nice explanation.

I would like to add one more variant of Case1:

Case I: one equation enough to solve

In some question, we have inherent integrality condition. In such cases, it may be possible to answer the question with one equation itself.

https://gmatclub.com/forum/max-has-125- ... 40249.html

Thanks.
Intern
Joined: 26 Dec 2016
Posts: 20
Re: Two variables - two equations trap  [#permalink]

### Show Tags

15 Feb 2017, 07:29
mikemcgarry wrote:
iliavko wrote:
Hi everyone!

I get confused when it comes to applying the "we need two different equations to solve for a variable" rule.

Can you please give an example when just having one eq with two variables is enough to solve?

Thank you!

Dear iliavko,

I'm happy to respond.

I will distinguish three cases:
Case I: one equation enough to solve
Case II: two equations needed to solve
Case III: two equations not enough to solve for two variables
I point out that Case II is statistically the most common case, and most often tested. That's the rule, but there are exceptions.

Case I only happens when the question is asking not for the value of x or the value of y, but simply for the value of some expression. Here's a very simple example. Suppose the question is "What is the value of y - x?" Suppose we are given the single equation 2y - x = 28 + x. We could rearrange that equation to solve for the desired expression, which equals 14, despite the fact that we cannot find the individual values of the variables.

Case II happens when we have two different equations that represent two lines with different slope. Any two lines in the x-y plane that have different slopes will intersect at exactly one and only one (x, y) point. That determines a unique and unambiguous solution.

Case III happens when the two lines have the same slope. For example:
Equation #1: x - 2y = 5
Equation #2: 4y + 6 = 2x
It's not necessarily obvious at first glance, but if you were to re-write these both in y = mx + b form, you would see that they have the same slope. If you try to solve them, you will get mathematical nonsense. In fact, they are parallel lines, and they never intersect, so there's no solution.
A variant on this case:
Equation #1: 3x + 2y = 5
Equation #2: 4x = 10 - 6y

This what mathematicians call a "degenerate system," because the two equations are really one equation in two different forms. Here, we can't solve, because absolutely every value that works in equation #1 will work in equation #2 as well, because they're the same equation! This has a continuous infinity of possible solutions, which in practice, is equivalent to having no solution.

Does all this make sense?
Mike

Hi,

thanks for your explanation. I think there is a small typo.
I guess the highlited part should be :
I) 3x + 2y = 5
II) 6x = 10 - 4y
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8804
Location: Pune, India
Re: Two variables - two equations trap  [#permalink]

### Show Tags

16 Feb 2017, 07:58
iliavko wrote:
Hi everyone!

I get confused when it comes to applying the "we need two different equations to solve for a variable" rule.

Can you please give an example when just having one eq with two variables is enough to solve?

Thank you!

Here I discuss two cases in which this happens:

https://www.veritasprep.com/blog/2011/0 ... -of-thumb/
_________________

Karishma
Veritas Prep GMAT Instructor

Re: Two variables - two equations trap &nbs [#permalink] 16 Feb 2017, 07:58
Display posts from previous: Sort by