The system of equations given below represen6y2u 3vcb4hsm6t: ppx ts 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 represents three planes in space. om ert go9 s+).i 9deouyvdvf-2 )f1sp9gd**k

$\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 space* x4lwaw ch:qtep7 *x4lql10s.

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

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

Consider the following s3c77y,qrdk mr2xy4y :jg o tis(z rz9/ystem 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 system of equations: lee,edtp0o2,

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

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 3,pyar9pamp75db);sb8fh) v oyh6amequations:

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

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 foevyqs e1z(f j7ua4f5/ llowing 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.