The system of equations given below ret/ z7j ,mzq;v/lldv2sswxq4 :presents three planes in space.

$\begin{aligned}
x+5 z & =2 \\
-2 x+y-6 z & =-1 \\
2 y+8 z & =6
\end{aligned}$

1. Show that this system of equations has an infinite number of solutions.
2. Find the parametric equations of the line of intersection of the three planes.

The system of equations given below representdc 0/v2y cnm:7bsdom 4f0pqc* s three planes in space.

$\begin{aligned}
x+2 y-2 z & =4 \\
3 x+5 y-4 z & =9 \quad[\text { where } a, b \in \mathbb{R}] \\
4 x+6 y+a z & =b
\end{aligned}
$

Find the set of values of a and b such that the three planes have no points of intersection.

The system of equations give8dmtdt v7)/-ozg 5iovn below represents three planes in space.

$\begin{aligned}
x-3 y+2 z & =5 \\
3 x+5 y+a z & =b \quad \text { [where } a, b \in \mathbb{R}] \\
4 x+2 y-3 z & =7
\end{aligned}$

Find the set of values of a and b such that the three planes have exactly one intersection point.

Consider the following system of equations: of3u wuvrys ,o:s:0 fj-os90 j

$\begin{aligned}
x-3 z & =-2 \\
-3 x+y+6 z & =3 \\
2 x-2 y+(a-4) z & =b-3
\end{aligned}$

where $a, b \in \mathbb{R}$ .
1. Find conditions on a and b for which
1. the system has no solutions;
2. the system has only one solution;
3. the system has an infinite number of solutions.
2. In the case where the number of solutions is infinite, find the general solution of the system of equations in Cartesian form.

Consider the following system 9toi+cflb+2b((.a5n5kn eybba 40 k cfydz+sof equations:

$\begin{aligned}
x+y+4 z & =1 \\
3 x+2 y+16 z & =5 \\
4 x+2 y+(a-1) z & =b-4
\end{aligned}$

where $a, b \in \mathbb{R}$.
1. Find conditions on a and b for which
1. the system has no solutions;
2. the system has only one solution;
3. the system has an infinite number of solutions.
2. In the case where the number of solutions is infinite, find the general solution of the system of equations in Cartesian form.

Consider the following sy p - e)onuuu-3kekyxv;g(3fd-stem of equations:

$\begin{aligned}
x+y+z & =-1 \\
4 x+2 y+z & =3 \\
9 x+3 y & =p
\end{aligned}$

where $p \in \mathbb{R} $.
1. Show that this system does not have a unique solution for any value of p .
2. 1. Determine the value of p for which the system is consistent.
2. For this value of p , find the general solution of the system.

Consider three planes represented bydf*)*ldni*c za/wv 1u the following system of equations:

$\begin{array}{c}
\Pi_{1}:-2 x+a y+4 z=2 \\
\Pi_{2}: x+b y-2 z=-1 \\
\Pi_{3}: 2 x-y+c z=3
\end{array}
$

Where $ a, b, c \in \mathbb{R}$.
1. State the values of a, b and c for which
1. The system has infinite solutions.
2. The system is inconsistent.
2. In the case where the system has infinite solutions, describe the geometric relationship between the three planes.
3. In the case where the system is inconsistent, identify one of the geometric relationships that could exist between the three planes.