Let the system of linear equations
$-x+2y-9z=7$
$-x+3y-7z=9$
$-2x+y+5z=8$
$-3x+y+13z=\lambda$
has a unique solution $x=\alpha, y=\beta, z=\gamma$. Then the distance of the point $(\alpha, \beta, \gamma)$ from the plane $2x-2y+z=\lambda$ is

If the system of linear equations
$2x + y - z = 3$
$x - y - z = \alpha$
$3x + 3y + \beta z = 3$
has infinitely many solutions,then $\alpha + \beta - \alpha \beta$ is equal to .... .

Give the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.
Statement $-1$ : If the graphs of two linear equations in two variables are neither parallel nor the same,then there is a unique solution to the system.
Statement $-2$ : If the system of equations $ax + by = 0, cx + dy = 0$ has a non-zero solution,then it has infinitely many solutions.
Statement $-3$ : The system $x + y + z = 1, x = y, y = 1 + z$ is inconsistent.
Statement $-4$ : If two of the equations in a system of three linear equations are inconsistent,then the whole system is inconsistent.

Let $\lambda \in R$. The system of linear equations
$2x_{1} - 4x_{2} + \lambda x_{3} = 1$
$x_{1} - 6x_{2} + x_{3} = 2$
$\lambda x_{1} - 10x_{2} + 4x_{3} = 3$
is inconsistent for:

The number of real values of $\lambda$ for which the system of linear equations $2x + 4y - \lambda z = 0$,$4x + \lambda y + 2z = 0$,and $\lambda x + 2y + 2z = 0$ has infinitely many solutions is:

If $(x, y, z)=(\alpha, \beta, \gamma)$ is the unique solution of the system of simultaneous linear equations $3x - 4y + z + 7 = 0$,$2x + 3y - z = 10$,and $x - 2y - 3z = 3$,then $\alpha = $