The system of equations given below represents three planes in space./glfi5jyy 93a c;(r)j ;qwypt65 zqls

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

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 repr p0m vw9 smsaoz*k-3nn do:z9,av,b) lesents 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 given below represents three planes in spag 6a7;3 krcos zz77lobce.

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

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

Consider the following system7vi 2c t2y1 im.gxbz*z of equations:

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

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 systc:n( inrco n2i s2de+4em of 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 system of equations:;1nyuzla3s3s3 z + i:3gmrpnq

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

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 representq7o33 q w1kyu/8o qivzed by 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.