Systems of Linear Equations

[Maximum mark: 6]

The system of equations given below represents three planes in space.

1. Show that this system of equations has an infinite number of solutions. [3]
   

2. Find the parametric equations of the line of intersection of the three planes. [3]
   

[Maximum mark: 9]

Consider the following system of equations:

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. [6]
      

2. In the case where the number of solutions is infinite, find the general
   solution of the system of equations in Cartesian form. [3]
   

[Maximum mark: 6]

The system of equations given below represents three planes in space.

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

[Maximum mark: 8]

Consider the following system of equations:

where $p \in \mathbb{R}$.

1. Show that this system does not have a unique solution for any value of $p$. [4]
   

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. [4]