roots

vipulgoyal wrote: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