Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations:

$3 x-y-z $$ =0 $, $-3 x+z $$ =0 $, $-3 x+2 y+z $$ =0 .$

Then the number of such points for which $x^2+y^2+z^2 \leq 100$ is

The following system of equation $3x – 2y + z = 0$, $\lambda x – 14y + 15z = 0$, $x + 2y – 3z = 0$ has a solution other than $x = y = z = 0$ for $\lambda $ equal to

Solve system of linear equations, using matrix method. $2 x-y=-2$ ; $3 x+4 y=3$

Examine the consistency of the system of equations. $x+y+z=1$ ; $2 x+3 y+2 z=2$ ; $a x+a y+2 a z=4$

Let $S$ be the set of all column matrices $\left[\begin{array}{l}b_1 \\ b_2 \\ b_3\end{array}\right]$ such that $b_1, b_2, b_3 \in R$ and the system of equations (in real variables)

$-x+2 y+5 z=b_1$

$2 x-4 y+3 z=b_2$

$x-2 y+2 z=b_3$

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each$\left[\begin{array}{l}b_1 \\ b_2 \\ b_3\end{array}\right]$ $\in$ $S$ ?

$(A)$ $x+2 y+3 z=b_1, 4 y+5 z=b_2$ and $x+2 y+6 z=b_3$

$(B)$ $x+y+3 z=b_1, 5 x+2 y+6 z=b_2$ and $-2 x-y-3 z=b_3$

$(C)$ $-x+2 y-5 z=b_1, 2 x-4 y+10 z=b_2$ and $x-2 y+5 z=b_3$

$(D)$ $x+2 y+5 z=b_1, 2 x+3 z=b_2$ and $x+4 y-5 z=b_3$