A system of two linear equations in two variables $x$ and $y$ is represented geometrically as two lines in the plane. These lines can intersect at a point, coincide, or be parallel as in the Figure 1.
Consider the system below, where $a$ and $b$ are constants
$$ \begin{align*} 3x -\ y &= 3 \\ ax + by &= 6 \end{align*} $$
(a) Find values for $a$ and $b$ for which the resulting system has a unique solution. Show that the solution is unique. Determine the solution.
(b) Find values for $a$ and $b$ for which the resulting system has infinitely many solutions. Show that linear system has infinitely many solutions. Determine the solution set.
(c) Find values for $a$ and $b$ for which the resulting system has no solution. Show that the linear system has no solution.
(d) Graph the lines for each of the linear systems in parts (a), (b), and (c) on separate graphs. Include your graphs on the same page with your answers to (a)-(c). Do not create your plots with pen and paper. You must create your plots using your favorite graphing software. I suggest 
GeoGebra$\textregistered$, or
Desmos$\textregistered$.
Consider a linear system with three linear equations in $x$, $y$, and $z$. The graph of each linear equation is a plane in a 3-dimensional coordinate system. In parts (a), (b), and (c) you are given a description of the geometry of the three planes and their graphs on the same coordinate system. From the geometrical description and the graphs create a system of three equations (one for each plane) in the three unknowns: $x$, $y$, and $z$.
(a) Find an example of such a linear system whose graph is given by Figure 2. All three planes intersect in a single line, the $x$-axis. Thus there are infinitely many solutions to the linear system: all of the points on the common line.
(b) Find an example of such a linear system whose graph is given by Figure 3. The three planes intersect at a single point, the origin. Thus the linear system has exactly one solution: the point that lies on all three planes.
(c) Find an example of such a linear system whose graph is given by Figure 4. Two of the planes are parallel and distinct. Yet the third plane is not parallel to them. Thus the pairwise intersection of the third plane with each parallel plane is a line, and these two lines are parallel to each other. Thus there are no solutions to the linear system because no point lies on all three planes simultaneously.
(d) Are there more different geometrical descriptions in which three planes can intersect? Create graphs of the other possible linear systems of three equations with three unknowns and include a graph of each. Include graphs as well as linear systems in your pdf for all of the remaining distinct geometrical descriptions. Do not create your plots with pen and paper. You must create your plots using your favorite graphing software. I suggest GeoGebra$\textregistered$, or
Desmos$\textregistered$.
