If the system of equations $3x - 2y + z = 0$,$\lambda x - 14y + 15z = 0$,and $x + 2y - 3z = 0$ has a solution other than $x = y = z = 0$,then $\lambda = $

If $(\alpha, \beta, \gamma)$ is the solution of the system of simultaneous linear equations given by $3x + 4y - 5z = -6$,$2x + 3y - 4z = -7$,and $4x - 2y + z = 9$,then find the value of $\alpha + 3\beta - 2\gamma$.

The system of equations $x + y + z = 6$,$x + 2y + 3z = 10$,and $x + 2y + \lambda z = \mu$ has no solution for:

If the system of equations
$2x + 7y + \lambda z = 3$
$3x + 2y + 5z = 4$
$x + \mu y + 32z = -1$
has infinitely many solutions,then $(\lambda - \mu)$ is equal to

Let $S$ be the set of all $\lambda \in \mathbb{R}$ for which the system of linear equations
$2x - y + 2z = 2$
$x - 2y + \lambda z = -4$
$x + \lambda y + z = 4$
has no solution. Then the set $S$

Let the system of equations $x+2y+3z=5$,$2x+3y+z=9$,and $4x+3y+\lambda z=\mu$ have an infinite number of solutions. Then $\lambda+2\mu$ is equal to: