Answer
See explanations below.
Work Step by Step
(a) Given $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$ being solutions of the system, we have
$a_1x_0+b_1y_0+c_1z_0=d_1$ ... $a_3x_1+b_3y_1+c_3z_1=d_3$. Test to see if $(\frac{x_0+x_1}{2}, \frac{y_0+y_1}{2}, \frac{z_0+z_1}{2} )$ is also a solution: $a_1(\frac{x_0+x_1}{2})+ b_1(\frac{y_0+y_1}{2})+ c_1(\frac{z_0+z_1}{2} )
=\frac{1}{2}(a_1x_0+b_1y_0+c_1z_0+a_1x_1+b_1y_1+c_1z_1)=\frac{1}{2}(d_1+d_1)=d_1$ which means that it is true.
Similarly we can test it for other equations to reach a conclusion that $(\frac{x_0+x_1}{2}, \frac{y_0+y_1}{2}, \frac{z_0+z_1}{2} )$ is also a solution to the system.
(b) Based on the results from above, if a system has two different solutions, we can always find a middle point between these two solutions which becomes a new solution, and use this new solution with one of the old solution to generate another middle point for another new solution. As this process can be repeated indefinitely, we can reach a conclusion that if the system has two different solutions, then it has infinitely many solutions.
