The system of equations given below represents three planesp1tlmffk ( n7f+l 7br8od+m( f 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 given8rc d g6*jz - t90jt7nti:xtay below represents 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 5:z::r lfa-g rdsz3 kzgiven 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 equatig1gyg6w ; m1/qi lb3vi8 yxvt4ons:

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

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 of equations: f3x rdf6+s0fq

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

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 equationey9ydfcue1*fo b .w*4s:

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

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 by the following sy l9//r ujk6 yrj 8o27delllpy(q, 4uelstem 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.