roots

[vipulgoyal]

If equation 3x-2y=5 and ax+6y=10 have no common root, a=?
(A) 6
(B) 3
(C) 0
(D) -3
(E) -9

[Brent@GMATPrepNow]

If equation 3x - 2y = 5 and ax + 6y = 10 have no common root, a=?
(A) 6
(B) 3
(C) 0
(D) -3
(E) -9

If two linear equations have no common root, then the variable portions of the equations are multiples of each other, BUT the constant (non-variable) part is not.

Some examples:
The following system of equations has no common root.
x + y = 3
2x + 2y = 5
Here, 2x + 2y is a multiple of x + y, since 2(x + y) = 2x + 2y
HOWEVER, (2)(3) does not equal 5

The following system of equations has no common root.
20x - 12y = 31
5x - 3y = 11
Here, 20x - 12y is a multiple of 5x - 3y, since 4(5x - 3y) = 20x - 12y
HOWEVER, (4)(11) does not equal 31

The following system of equations has no common root.
7x - 10y = 5
-14x + 20y = 19
Here, -14x + 20y is a multiple of 7x - 10y, since -2(7x - 10y) = -14x + 20y
HOWEVER, (-2)(5) does not equal 19


Okay, onto the question . . .
3x - 2y = 5
ax + 6y = 10
Notice that, if we multiply -2y by -3, we get +6y
So, let's multiply 3x by -3 as well . . .
We get: -9x + 6y = 10

As we can see, -9x + 6y is a multiple of 3x - 2y, since -3(3x - 2y) = -9x + 6y
HOWEVER, (-3)(5) does not equal 10
So, the following system does not have a common root:
3x - 2y = 5
-9x + 6y = 10

Answer: E

Cheers,
Brent

[ollapodrida]

I'm assuming what you mean when you say that the two equations "have no common root" is that they cannot be solved simultaneously.

In that case, I would pick (E), a=-9, and here's why:

If a=-9,

We have the following two equations:
3x-2y = 5
-9x+6y=10
If you multiply the first equation by 3 and keep the second equation as it is, you get
9x-6y=15
-9x+6y=10
--------------
Adding the two equations above gives
0+0=25 --> Not possible

In general, if you have a question that might require trying out different answer choices, it is best to start from the bottom. In this case, all the other values of a will produce equations that can be solved simultaneously.

[mohit_1607]

If equation 3x-2y=5 and ax+6y=10 have no common root, a=?
(A) 6
(B) 3
(C) 0
(D) -3
(E) -9

Explanation 1:
If the equations have no common root, then they must be parallel.
For || lines, slopes are equal-->m1=m2
Therefore, 3/2 = -a/6 -> a = -9
Option E

Explanation 2:
For 2 linear equations: a1x + b1y = c1 and a2x + b2y = c2
The eqns have No soln if a1/a2 = b1/b2 != c1/c2
This gives 3/a = -2/6 != 5/10
--> a = -9
Option E