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It is given that there is no solution to the system x+2y=3,ax+by=4.Which of the following is true?

a) a has a unique value
b) b has a unique value
c) a can have more than one value
d) a has exactly two different values.
Explain the result



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X + 2 y = 3
a x+ by = 4

If b ≠ 2a, then  there is exactly one solution.

Let a = k  and b = 2 k =>   the system of equations becomes:
         x+2y = 3   and   x+2y = 4/k 

They will not have any solution if a = k and b = 2k ,  with  k ≠ 4/3.  That is if a=1, b=2, then there is no solution.  For a=2, b=4 there is no solution..  for a=3, b=6 there is no solution.

If k = 4/3, or a= 4/3 and  b=8/3,  then there are infinite number of solutions.

   In that case of a=k, and b=2k,  we get
            x+2y=3  and   kx+2ky=4  or   x+2y = 4/k  =>  k must be 3. 
Another way,
         matrix  [ 1 2 ]  should be singular. 
                    [ a b ] 

1 4 1
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