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

 It is currently 21 Apr 2019, 13:19

### 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

# How many ordered pairs of real numbers (x,y) satisfy the following sys

Author Message
TAGS:

### Hide Tags

Manager
Status: Manager
Joined: 02 Nov 2018
Posts: 179
How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

18 Mar 2019, 08:58
10
00:00

Difficulty:

95% (hard)

Question Stats:

26% (02:28) correct 74% (02:17) wrong based on 85 sessions

### HideShow timer Statistics

How many ordered pairs of real numbers$$(x,y)$$ satisfy the following system of
equations?

$$x + 3y = 3$$

$$||x| − |y||$$= 1

(A) 1

(B) 2

(C) 3

(D) 4

(E) 8

_________________
Give a kudos if u find my post helpful. kudos motivates active discussions

Intern
Joined: 02 Mar 2019
Posts: 3
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

08 Apr 2019, 11:18
5
Detailed Solution:

1. Solve two equations (Solving by Combination:

x+3y=3
||x|−|y||= 1

Here for the second equation, give values for x and y

Case 1: x=+x and y=+y
x+3y=3
x-y = 1
Solve the equation
x=1.5 and y=0.5

Case 2: x=-x and y=-y
x+3y=3
-x+y = 1
Solve the equation
x=0 and y=1

Case 3: x=-x and y=y
x+3y=3
-x-y = 1
Solve the equation
x=-3 and y=2

Case 4: x=+x and y=-y
Try yourself, you will get one of the same solution above

Hence we have three solution pairs -> (x,y) = (1.5,0.5), (0,1) and (-3,2)

HENCE SOLVED.
##### General Discussion
Intern
Joined: 24 Jul 2013
Posts: 29
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

18 Mar 2019, 10:20
How many ordered pairs of real numbers$$(x,y)$$ satisfy the following system of
equations?

$$x + 3y = 3$$

$$||x| − |y||$$= 1

(A) 1

(B) 2

(C) 3

(D) 4

(E) 8

Is it C? I am getting 3 points (-3,2), (0,1) and (1.5,0,5)
Manager
Status: Manager
Joined: 02 Nov 2018
Posts: 179
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

18 Mar 2019, 10:36
1
hi longranger25
_________________
Give a kudos if u find my post helpful. kudos motivates active discussions

Director
Joined: 27 May 2012
Posts: 731
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

21 Mar 2019, 13:55
1
longranger25 wrote:
How many ordered pairs of real numbers$$(x,y)$$ satisfy the following system of
equations?

$$x + 3y = 3$$

$$||x| − |y||$$= 1

(A) 1

(B) 2

(C) 3

(D) 4

(E) 8

Is it C? I am getting 3 points (-3,2), (0,1) and (1.5,0,5)

_________________
- Stne
Manager
Joined: 19 Sep 2017
Posts: 100
Location: United Kingdom
GPA: 3.9
WE: Account Management (Other)
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

06 Apr 2019, 07:36
Hi Experts,
chetan2u Bunuel VeritasKarishma Is there a better way or formula to find ordered pairs of such equations?
_________________
Cheers!!

~If you do what's easy, your life will be hard. If you do what's hard, your life will be easy~
Intern
Joined: 07 Apr 2019
Posts: 3
How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

08 Apr 2019, 17:36
How many ordered pairs of real numbers$$(x,y)$$ satisfy the following system of
equations?

\begin{align}x + 3y &= 3 \quad (1)\\ ||x| − |y|| &= 1\quad (2)\end{align}

(A) 1 (B) 2 (C) 3 (D) 4 (E) 8

One can drop an absolute value by introducing cases, e.g. $$\lvert x -7\rvert = \begin{cases}\phantom{-(} x -7&,\text{if } x \geq 7 \\ -(x -7) &, \text{if } 7 > x\,. \end{cases}$$

Depending on whether the argument is positive or negative (or zero), we have to apply a change of sign. Dropping all three absolute values in equation (2) results in expressions of the form $$\pm (\pm x \pm y) = 1$$ (all combinations possible), which can be written as cases

$$x = \begin{cases}\phantom{-} y +1 &, C_1(x,y) \\ \phantom{-} y -1 &, C_2(x,y) \\ -y +1 &, C_3(x,y) \\ -y -1 &, C_4(x,y) \end{cases}$$

where I should (but don't) keep track of the conditions imposed on $$x$$ and $$y$$ in the four cases. The cases represent straight (half) lines in the plane, none of which is parallel to the line of equation (1). Therefore, each of the four lines has exactly one point in common with (1). The points turn out to be $$(-3,2), (0,1), (1.5,0.5)$$.
Since I didn't keep track of the conditions $$C_k$$, I have to check if the above pairs solve equation (2), they do. The correct answer is (C).
Intern
Joined: 05 Mar 2018
Posts: 20
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

09 Apr 2019, 08:15
UranousCold wrote:
Detailed Solution:

1. Solve two equations (Solving by Combination:

x+3y=3
||x|−|y||= 1

Here for the second equation, give values for x and y

Case 1: x=+x and y=+y
x+3y=3
x-y = 1
Solve the equation
x=1.5 and y=0.5

Case 2: x=-x and y=-y
x+3y=3
-x+y = 1
Solve the equation
x=0 and y=1

Case 3: x=-x and y=y
x+3y=3
-x-y = 1
Solve the equation
x=-3 and y=2

Case 4: x=+x and y=-y
Try yourself, you will get one of the same solution above

Hence we have three solution pairs -> (x,y) = (1.5,0.5), (0,1) and (-3,2)

HENCE SOLVED.

Why have we not considered the outer modulus while solving? x and y may be +ve or -ve, but what about the outermost mod? considering that would make the solution lengthier and more time consuming.
Intern
Joined: 02 Mar 2019
Posts: 3
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

09 Apr 2019, 19:27
MeBossBaby : Thanks for the question.

Absolute value equations will generally have two solutions.
For Example:
|x| could be +2 or -2 as the absolute value of |x| here simply refers how far number +2 or -2 is away from zero in the number line.
This means we have two solutions, positive and negative.

Even If I say x with double absolute value ||x|| and apply the two values +2 and -2, I will get the same solution.
||+2|| = +2 and ||-2|| = +2, this mean x can have still hold two values +2 and -2.

We can apply this same principle on the two variable equation ||x|−|y||= 1, and can safely say it can have four solutions +x, -x, +y and -y.
Intern
Joined: 02 Mar 2019
Posts: 3
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

09 Apr 2019, 20:14
MeBossBaby wrote:
UranousCold wrote:
Detailed Solution:

1. Solve two equations (Solving by Combination:

x+3y=3
||x|−|y||= 1

Here for the second equation, give values for x and y

Case 1: x=+x and y=+y
x+3y=3
x-y = 1
Solve the equation
x=1.5 and y=0.5

Case 2: x=-x and y=-y
x+3y=3
-x+y = 1
Solve the equation
x=0 and y=1

Case 3: x=-x and y=y
x+3y=3
-x-y = 1
Solve the equation
x=-3 and y=2

Case 4: x=+x and y=-y
Try yourself, you will get one of the same solution above

Hence we have three solution pairs -> (x,y) = (1.5,0.5), (0,1) and (-3,2)

HENCE SOLVED.

Why have we not considered the outer modulus while solving? x and y may be +ve or -ve, but what about the outermost mod? considering that would make the solution lengthier and more time consuming.

MeBossBaby : Thanks for the question.

Absolute value equations will generally have two solutions.
For Example:
|x| could be +2 or -2 as the absolute value of |x| here simply refers how far number +2 or -2 is away from zero in the number line.
This means we have two solutions, positive and negative.

Even If I say x with double absolute value ||x|| and apply the two values +2 and -2, I will get the same solution.
||+2|| = +2 and ||-2|| = +2, this mean x can have still hold two values +2 and -2.

We can apply this same principle on the two variable equation ||x|−|y||= 1, and can safely say it can have four solutions +x, -x, +y and -y.
Intern
Joined: 05 Mar 2018
Posts: 20
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys  [#permalink]

### Show Tags

11 Apr 2019, 00:32
1
Okay.... I think I get it now..
Thanks UranousCold!
Re: How many ordered pairs of real numbers (x,y) satisfy the following sys   [#permalink] 11 Apr 2019, 00:32
Display posts from previous: Sort by