The system of equations has how many solutions?

[GMATprofessional21]

The system of equations has how many solutions?

3x − 6y = 9

2y − x − 3 = 0

A None
B Exactly 1
C Exactly 2
D Exactly 3
E Infinitely many

[Brent@GMATPrepNow]

The system of equations has how many solutions?

3x − 6y = 9

2y − x − 3 = 0

A None
B Exactly 1
C Exactly 2
D Exactly 3
E Infinitely many

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

Take 2x = 2y + 6 and divide both sides by 2 to get an EQUIVALENT equation x = y + 3
Now subtract y from both sides to get: to get another EQUIVALENT equation x - y = 3
Since this equation (x - y = 3) is identical to the other equation x - y = 3, we can be certain that ANY solution to x - y = 3 will also be a solution to x - y = 3

Since there are infinitely many solutions to x - y = 3, there will be infinitely many solutions to the entire SYSTEM.

Answer: E

Cheers,
Brent

[Brent@GMATPrepNow]

The system of equations has how many solutions?

3x − 6y = 9

2y − x − 3 = 0

A None
B Exactly 1
C Exactly 2
D Exactly 3
E Infinitely many

Another 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

[Scott@TargetTestPrep]

The system of equations has how many solutions?

3x − 6y = 9

2y − x − 3 = 0

A None
B Exactly 1
C Exactly 2
D Exactly 3
E Infinitely many

Solution:

Multiplying the second equation by 3, we have:

6y - 3x - 9 = 0

6y - 3x = 9

-3x + 6y = 9

Adding the equations together, we have:

0 = 18

Since 0 can’t be equal to 18, there are no solutions that satisfy the given system of equations.

Answer: A