QJEEM23Ma23d8: JEE Main 2023 Mathematics - Class 12 JEEM23Ma23d8 Solution: If the system of equations has infinitely many solutions, then…

Question

Verified Solution • 2025–26 Session moderate

If the system of equations

$x+y+a z=b$

$2 x+5 y+2 z=6$

$x+2 y+3 z=3$

has infinitely many solutions, then $2 a+3 b$ is equal to :

Accepted Answer

Given system of equations,

\text { and } \quad \begin{aligned}x+y+a z &amp; =b \\2 x+5 y+2 z &amp; =6 \\x+2 y+3 z &amp; =3\end{aligned}

Since, given system of equation has infinitely many solutions

\therefore D=0 \text { and } D_1=D_2=D_3=0

\begin{aligned}&amp; \text { Here, } D=\left|\begin{array}{lll}1 &amp; 1 &amp; a \\2 &amp; 5 &amp; 2 \\1 &amp; 2 &amp; 3\end{array}\right|=0 \\\\&amp; \Rightarrow 1(15-4)-1(6-2)+a(4-5)=0 \\\\&amp; \Rightarrow -a+11-4=0 \\\\&amp; \Rightarrow a=7\end{aligned}

\begin{aligned}&amp; \text { and } D_1=\left|\begin{array}{ccc}b &amp; 1 &amp; a \\6 &amp; 5 &amp; 2 \\3 &amp; 2 &amp; 3\end{array}\right|=0 \\\\&amp; \Rightarrow b(15-4)-1(18-6)+a(12-15)=0 \\\\&amp; \Rightarrow 11 b-12-3 a=0 \\\\&amp; \Rightarrow 11 b-12-21=0 ~~~~~~~(\because a=7)\\\\&amp; \Rightarrow 11 b-33=0 \\\\&amp; \Rightarrow b=3 \\\\&amp; \therefore 2 a+3 b=2(7)+3(3)=14+9=23\end{aligned}

Step-by-Step Verified Solution