JEE Main 2026 — Mathematics Algebra

For the given system of equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, $\Delta = 0$Δ=0.

$$\Delta = \begin{vmatrix} 1 & 5 & 6 \\ 2 & 3 & 4 \\ 1 & 6 & a \end{vmatrix} = 0$$Δ=​121​536​64a​​=0

Expanding along the first row:

$$1(3a - 24) - 5(2a - 4) + 6(12 - 3) = 0$$1(3a−24)−5(2a−4)+6(12−3)=0

$$3a - 24 - 10a + 20 + 54 = 0$$3a−24−10a+20+54=0

$$-7a + 50 = 0 \Rightarrow a = \dfrac{50}{7}$$−7a+50=0⇒a=750​

For infinitely many solutions, we must also have $\Delta_z = 0$Δz​=0.

$$\Delta_z = \begin{vmatrix} 1 & 5 & 4 \\ 2 & 3 & 7 \\ 1 & 6 & b \end{vmatrix} = 0$$Δz​=​121​536​47b​​=0

Expanding along the first row:

$$1(3b - 42) - 5(2b - 7) + 4(12 - 3) = 0$$1(3b−42)−5(2b−7)+4(12−3)=0

$$3b - 42 - 10b + 35 + 36 = 0$$3b−42−10b+35+36=0

$$-7b + 29 = 0 \Rightarrow b = \dfrac{29}{7}$$−7b+29=0⇒b=729​

The point $(a, b)$(a,b) is $\left(\dfrac{50}{7}, \dfrac{29}{7}\right)$(750​,729​).

Checking the given options, we evaluate $a - b$a−b:

$$a - b = \dfrac{50}{7} - \dfrac{29}{7} = \dfrac{21}{7} = 3$$a−b=750​−729​=721​=3

Therefore, the point $(a, b)$(a,b) lies on the line $x - y = 3$x−y=3.

Answer: $x - y = 3$x−y=3