Question

The system of equations for the variables x and y
a x+b y = e
c x + d y = f
has a unique solution only if
(A) a d − b c ≠ 0 (B) a c − b d ≠ 0 (C) a + c ≠ b + d (D) a − c ≠ b − d

Answer 1

In order to have unique solution, the straight lines represented by the equations should be intersecting (not coinciding or parallel)

So, slopes should not be equal

i.e -a/b ≠ -c/d

⇒ ad≠bc

⇒ad-bc≠0 (option A)

Answer 2

Option(A) is the correct answer.

Given,

Equation 1 : a x+b y = e

Equation 2 : cx + dy = f

To Find,

The relation between a,b,c and d if the system of equations has a unique solution =?

Solution,

We know that the system of equations has a unique solution only if

(a1/a2) ≠ (b1/b2) ≠ (c1/c2)

Putting the values according to the given system of equations, we get

(a/c) ≠ (b/d) ≠ (e/f)

(a/c) ≠ (b/d)

(a/c) - (b/d)   ≠  0

(ad - bc) / dc  ≠  0

ad - bc  ≠  0

Hence, the system of equations has a unique solution only if ad−bc≠0.