QJEEM23Mac096: JEE Main 2023 Mathematics - Class 12 JEEM23Mac096 Solution: Let the system of linear equations x + y + k z = 2 2 x + 3 y −…

Question

Verified Solution • 2025–26 Session moderate

Let the system of linear equations

$x + y + k z = 2$

$2 x + 3 y - z = 1$

$3 x + 4 y + 2 z = k$

have infinitely many solutions. Then the system

$(k + 1) x + (2 k - 1) y = 7$

$(2 k + 1) x + (k + 5) y = 10$

has :

Accepted Answer

$x + y + k z = 2$ ............(i)

$2 x + 3 y - z = 1$ ..........(ii)

$3 x + 4 y + 2 z = k$ ......(iii)

(1) + (2)

$3 x + 4 y + z (k - 1) = 3$

Comparing with (3)

$k = 3$

Now, $4 x + 5 y = 7$

$\Rightarrow 3 x + 3 y = 3$

$7 x + 8 y = 10$

as $\frac{4}{7} \neq \frac{5}{8}$

$∴$ Unique solution satisfying $x + y = 1$

Step-by-Step Verified Solution