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# The system of equations above has how many solutions?

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Manager
Joined: 11 May 2007
Posts: 103
The system of equations above has how many solutions?  [#permalink]

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Updated on: 06 Jan 2014, 03:07
3
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Difficulty:

25% (medium)

Question Stats:

56% (00:28) correct 44% (00:26) wrong based on 469 sessions

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x - y = 3
2x = 2y + 6

The system of equations above has how many solutions?

(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

Originally posted by yogachgolf on 15 Nov 2007, 14:21.
Last edited by Bunuel on 06 Jan 2014, 03:07, edited 1 time in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
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07 Jul 2011, 21:41
1
6
ssarkar wrote:
x-y=3
2x=2y+6

The system of equations above has how many solutions?
(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

Say, the given equations are:

ax + by = c
dx + ey = f

If $$\frac{a}{d} \neq \frac{b}{e}$$, then the system of equations has a unique solution.

If $$\frac{a}{d} = \frac{b}{e} \neq \frac{c}{f}$$, then the system of equations has no solution.

$$\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$$, then the system of equations has infinitely many solutions.

Here, $$\frac{1}{2} = \frac{-1}{-2} = \frac{3}{6}$$ so there are infinitely many solutions.
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04 Apr 2011, 04:57
4
1
petrifiedbutstanding wrote:
Attachment:
1.JPG

The system of equations above has how many solutions?
(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

Sol:

$$x-y=3$$ is same as $$2x=2y+6$$

$$2x=2y+6$$
Dividing both sides by 2;
$$x=y+3$$
Subtracting y from both sides;
$$x-y=3$$

Thus, we have only one equation:
$$x-y=3$$
This has infinitely many solutions such as:
x=3,y=0
x=100,y=97
x=-100,y=-103
x=0.001, y=-2.999
x=1, y=-2
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Manager
Joined: 08 Nov 2007
Posts: 93

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15 Nov 2007, 14:52
yogachgolf wrote:
yogachgolf wrote:
x-y = 3
2x= 2y+6

The system of equations above has how many
solutions?

(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

I thought it's A as well. But OA is E?

I see your point ... A + B can be an infinite number of things... I suppose we answered it on the basis that the two solutions given gave us no CERTAIN solutions...

I think the question is a touch ambiguous, but I suppose we could have read it more closely.
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Joined: 29 Aug 2007
Posts: 2410

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15 Nov 2007, 15:19
alrussell wrote:
yogachgolf wrote:
yogachgolf wrote:
x-y = 3
2x= 2y+6

The system of equations above has how many solutions?

(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

I thought it's A as well. But OA is E?

I see your point ... A + B can be an infinite number of things... I suppose we answered it on the basis that the two solutions given gave us no CERTAIN solutions...

I think the question is a touch ambiguous, but I suppose we could have read it more closely.

seems ok but we cannot solve the equations do not provide any value for x and y.
Intern
Joined: 06 Sep 2007
Posts: 8

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Updated on: 16 Nov 2007, 22:42
to satisfy x-y=3

(1,-2)
(2,-1)
......
infinitely

two equation are same.

If we think these based on function, these are same linear.

Originally posted by LEE SANG IL on 15 Nov 2007, 23:12.
Last edited by LEE SANG IL on 16 Nov 2007, 22:42, edited 1 time in total.
Manager
Joined: 25 Jul 2007
Posts: 107

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16 Nov 2007, 02:50
Any single linear equation with more than 1 variable in it has infinite solutions (provided no constraints are given).

Edit: I do not see any ambiguity in the question.

The number of solutions for a linear equation is the number of possible values the variables can have so as to satisfy the equation. There are infinite possible values for the variables x & y in the given equation and therefore there are infinite solutions.
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04 Apr 2011, 05:50
1
Both the equations are same, so it will have infinitely many solutions.

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06 Jul 2011, 19:53
1
given two equations are exactly same.

so different values of x , will yield different values of y.

=> infinite solutions

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07 Jul 2011, 10:47
ssarkar wrote:
x-y=3
2x=2y+6

The system of equations above has how many solutions?
(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

x-y=3 ---------------1
2x=2y+6 ---------------2

Divide equation 2 by 2:
2x/2=(2y+6)/2
x=y+3
x-y=3----------------3

Equation 1 and 3 are equal and thus have infinitely many solutions:

x-y=3
x=5, y=2
x=6, y=3
x=7, y=4

Ans: "E"
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Re: Pls help me to solve this problem.  [#permalink]

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31 Jul 2011, 05:19
tracyyahoo wrote:
x-y=3
2x=2y+6

The system of equations above has how many solutions?

a) None
b) Exactly one
c) Exactly two
d) Exactly three
e) Infinitely many

I calculate that x will substracted by the equation, and I think it is b.

Actually both the equations represent the same i.e.

2x = 2y+6
=> x=y+3
=> x-y = 3 same as eq 1

Hence there are Infinitely many solutions to the equations as there are no restrictions

Hence E
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Joined: 20 Dec 2013
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Re: x-y = 3 2x= 2y+6 The system of equations above has how many  [#permalink]

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05 Jan 2014, 23:53
yogachgolf wrote:
x-y = 3
2x= 2y+6

The system of equations above has how many
solutions?

(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

Equation 1 = x = y + 3
Equation 2 = x = y + 3

Since both the equations represent a single line hence there will be infinitely many solutions for this.
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Posts: 118
Re: The system of equations above has how many solutions?  [#permalink]

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22 Aug 2015, 06:35
yogachgolf wrote:
x - y = 3
2x = 2y + 6

The system of equations above has how many solutions?

(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

Note that they are not parallel but one and the same. Therefore, x and y take infinitely many values.
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Re: The system of equations above has how many solutions?  [#permalink]

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19 Sep 2016, 09:46
ax + by = c
dx + ey = f

Very useful for line problems. This question was too easy. E is the right choice.
Director
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Posts: 590
Schools: Cambridge'16
Re: The system of equations above has how many solutions?  [#permalink]

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21 Sep 2016, 00:30
20 sec approach

x=y+3
2x=2y+6

the second equation is the first one doubled, so it is the same equation

E
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Posts: 2889
Re: The system of equations above has how many solutions?  [#permalink]

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10 May 2018, 09:19
Top Contributor
yogachgolf wrote:
x - y = 3
2x = 2y + 6

The system of equations above has how many solutions?

(A) None
(B) Exactly one
(C) Exactly two
(D) Exactly three
(E) Infinitely many

One approach is to begin solving the system (using the elimination method) and see what happens.

Given:
x - y = 3
2x = 2y + 6

Take bottom equation and divide both sides by 2 to get:
x - y = 3
x = y + 3

Take bottom equation and subtract y from both sides to get:
x - y = 3
x - y = 3

Now subtract the bottom equation from the top equation to get:
0x + 0y = 0
As we can see, this equation has infinitely many solutions.

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Re: The system of equations above has how many solutions? &nbs [#permalink] 10 May 2018, 09:19
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