The system of equations 2x + y = 5; x - 3y = -1 and 3x + 4y = k is consistent, then k is equal to :

Given: 

2x + y = 5; x - 3y = -1 and 3x + 4y = k is consistent

Concept used: 

if the system is consistent then Rank of A (Coefficient Matrix) = Rank of (A: B) Augmented Matrix = n (Number of Variables)

Calculations:

\(\left|\begin{array}{ccc} 2 & 1 & 5 \\ 1 & -3 & -1 \\ 3 & 4 & k \end{array}\right|\)

⇒ R₂  = 2R2 - R1

⇒ R₃ = 2R3 - 3R₁

\(\left|\begin{array}{ccc} 2 & 1 & 5 \\ 0 & -7 & -7 \\ 0 & 5 & 2k - 15 \end{array}\right|\)

⇒  R3 = 7R₃ + 5R₂

\(\left|\begin{array}{ccc} 2 & 1 & 5 \\ 0 & -7 & -7 \\ 0 & 0 & 14k - 140 \end{array}\right|\)

14k - 140 = 0 for rank of A and A:B = Number of variables i.e 2 

∴ k = 10 

Hence, the correct option is 4.