If $x=\alpha , y=\beta , z=y$ is the unique solution of the system of equations $5x-7y+3z=0,$ $7x+10y-8z=3$ and $2x+3y-4z+4=0,$ then $\beta =$

Given equations are

$5x-2y+3z=0 ...\left(1\right)$

$7x+10y-8z=3 ....\left(2\right)$

$2x+3y-4z=-4 ...\left(3\right)$

Now,

$D=\left|\begin{array}{ccc}5& -2& 3\\ 7& 10& -8\\ 2& 3& -4\end{array}\right|$

$\therefore D=5\left(-40-\left(-8\right)\times 3\right)+2\left(-28+16\right)+3\left(21-20\right)$

$\therefore D=-80-24+3=-101$

$D_y=\left|\begin{array}{ccc}5& 0& 3\\ 7& 3& -8\\ 2& -4& 4\end{array}\right|$

$\therefore D_y=5\left(12-32\right)+3\left(-28-6\right)=-202$

$\therefore y=\beta =\frac{D_y}{D}=\frac{-202}{-101}=2$

$\therefore \beta =2$