The system of equations above has how many solutions?

Question

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

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

KarishmaB · Accepted Answer

Say, the given equations are:

ax + by = c
dx + ey = f

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

If  \frac{a}{d} = \frac{b}{e} 
eq \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.

fluke · Answer

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

BrentGMATPrepNow · Answer

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

alrussell · Answer

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.

GMAT TIGER · Answer

seems ok but we cannot solve the equations do not provide any value for x and y.

LEE SANG IL · Answer

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

jbs · Answer

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.

subhashghosh · Answer

Both the equations are same, so it will have infinitely many solutions.

Answer - E.

Spidy001 · Answer

given two equations are exactly same.

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

=> infinite solutions

Answer is E.

fluke · Answer

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"

Sudhanshuacharya · Answer

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

PerfectScores · Answer

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.

mathivanan · Answer

Note that they are not parallel but one and the same. Therefore, x and y take infinitely many values.

utkarshthapak · Answer

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.

Temurkhon · Answer

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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The system of equations above has how many solutions?

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