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The system of equations above has how many solutions?

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yogachgolf
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The system of equations above has how many solutions? [#permalink] New post   Updated on: 06 Jan 2014, 03:07
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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
ShowHide Answer
Official Answer
E

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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KarishmaB
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Re: algebra [#permalink] New post   07 Jul 2011, 21:41
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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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fluke
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Re: System of Equations [#permalink] New post   04 Apr 2011, 04:57
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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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BrentGMATPrepNow
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The system of equations above has how many solutions? [#permalink] New post   Updated on: 05 Mar 2021, 10:40
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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.

Answer: E

Originally posted by BrentGMATPrepNow on 10 May 2018, 09:19.
Last edited by BrentGMATPrepNow on 05 Mar 2021, 10:40, edited 1 time in total.
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alrussell
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Re: Equations & Solutions [#permalink] New post   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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Re: Equations & Solutions [#permalink] New post   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.
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OA is E [#permalink] New post   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.

therefore answer is E

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.
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[#permalink] New post   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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Re: System of Equations [#permalink] New post   04 Apr 2011, 05:50
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Both the equations are same, so it will have infinitely many solutions.

Answer - E.
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Re: algebra [#permalink] New post   06 Jul 2011, 19:53
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given two equations are exactly same.

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

=> infinite solutions

Answer is E.
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Re: algebra [#permalink] New post   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] New post   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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Re: x-y = 3 2x= 2y+6 The system of equations above has how many [#permalink] New post   05 Jan 2014, 23:53
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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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Re: The system of equations above has how many solutions? [#permalink] New post   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] New post   19 Sep 2016, 09:46
ax + by = c
dx + ey = f

If ad≠bead≠be, then the system of equations has a unique solution.

If ad=be≠cfad=be≠cf, then the system of equations has no solution.

ad=be=cfad=be=cf, then the system of equations has infinitely many solutions.

Very useful for line problems. This question was too easy. E is the right choice.
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Re: The system of equations above has how many solutions? [#permalink] New post   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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Re: The system of equations above has how many solutions? [#permalink] New post   20 Jan 2024, 05:29
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