Question

Solve the system of equations
$x+3y-2z=02x-y+4z=0x-11y+14z=0$
then find the value of $2(|y+z)/x|$

Answer 1

We have
$x+3y-2z=02x-y+4z=0x-11y+14z=0$
The given system of equations in the matrix form are written as below:
$\left[\begin{array}{ccc}1& 3& -2\\ 2& -1& 4\\ 1& -11& 14\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
$AX=0...\left(1\right)$
where
$A=\left[\begin{array}{ccc}1& 3& -2\\ 2& -1& 4\\ 1& -11& 14\end{array}\right];X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $0=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
$\therefore |A|=1(-14+44)-3(28-4)-2(-22+1)=30-72+42=0$
and therefore the system has a non-trivial solution. Now, we may write first two of the given equations
$x+3y=2z$ and $2x-y=-4z$
Solving these equations in terms of $z$, we get
$x=-\frac{10}{7}z$ and $y=\frac{8}{7}z$
Now if $z=7k$, then $x=-10k$ and $y=8k$
Hence, $x=-10k,y=8k$ and $z=7k$ where $k$ is arbitrary, are the required solutions.
$\Rightarrow 2(|y+z)/x|=3$