For real numbers $\alpha$ and $\beta$,consider the following system of linear equations:
$x+y-z=2, x+2y+\alpha z=1, 2x-y+z=\beta$. If the system has infinite solutions,then $\alpha+\beta$ is equal to $.....$

If $S$ is the set of distinct values of $b$ for which the following system of linear equations $x + y + z = 1$,$x + ay + z = 1$,and $ax + by + z = 0$ has no solution,then $S$ is:

For how many different values of $a$ does the following system of linear equations have at least two distinct solutions?
$ax + y = 0$
$x + (a + 10)y = 0$

The values of $a$ and $b$,for which the system of equations $2x + 3y + 6z = 8$,$x + 2y + az = 5$,and $3x + 5y + 9z = b$ has no solution,are:

Given $A = \begin{bmatrix} 1 & 3 \\ 2 & 2 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If $A - \lambda I$ is a singular matrix,then:

Let the system of linear equations $x+y+\alpha z=2$,$3x+y+z=4$,and $x+2z=1$ have a unique solution $(x^{*}, y^{*}, z^{*})$. If $(\alpha, x^{*}), (y^{*}, \alpha)$ and $(x^{*}, -y^{*})$ are collinear points,then the sum of absolute values of all possible values of $\alpha$ is