# Find the condition so that system of equation is consistent

Consider the system of equations :

$$x + y +5z = 3$$

$$x + 2y + 4z = k$$

$$x + 2y + mz = 5$$ then this system of equation is consistent if

(a) $$m \ne 4$$

(b) $$k \ne 5$$

(c) $$m = 4$$

(d) $$k =5$$

Reducing, this system by subtracting equations (2) and (3) I get :

$$x + y + 5z = 3$$

$$x + 2y + 4z = k$$

$$(m -4)z = 5 -k$$

Now, the given system is consistent if $$m \ne 4$$,so only condition I require for consistency is $$m \ne 4$$

but if I take $$k = 5$$, then again the system is consistent

It will have unique solutions if $$m \ne 4$$

and infinite solution if $$m = 4$$

So, my question is are both (a) and (d) the correct choices for this question ?

Or is only (a) correct ?

Thank you.

If $$k=5$$ the system is consistent for all values of $$m$$