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 solutio…

Question

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,

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

Since, given system of equation has infinitely many solutions

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

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

\]

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

\]

Solutions · 1