For what value of k does the system of equations

x + 2y = 3, 5x + ky + 7 = 0


have (i) a unique solution, (ii) no solution?


Also, show that there is no value of k for which the given system of equations has infinitely many solutions.

(i) Given: x + 2y = 3 – eq 1


5x + ky + 7 = 0 – eq 2


Here,


a1 = 1, b1 = 2, c1 = - 3


a2 = 5, b2 = k, c2 = 7


Given systems of equations has a unique solution


∴ ≠


≠


k ≠10


∴ k ≠ 10


(ii) Given: x + 2y = 3 – eq 1


5x + ky + 7 = 0 – eq 2


Here,


a1 = 1, b1 = 2, c1 = - 3


a2 = 5, b2 = k, c2 = 7


Given that system of equations has no solution


∴ = ≠


Here,


=


Here,


k = 10


∴ k = 10


For the system of equations to have infinitely many solutions


= =


= = which is wrong.


That is, for any value of k the give system of equations cannot have infinitely many solutions.


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