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Math 511: Linear Algebra

Graphing Linear Equations

Graphing Linear Equations¶

You can represent a system of two linear equations in two variables $x$ and $y$ geometrically as two lines in the plane. These lines can intersect at a point, coincide, or be parallel as in the Figure 1.


Three plots of a linear system of two equations on the plane.
Figure 1

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.

(c) Find values for $a$ and $b$ for which the resulting 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 in this cell or add a new code cell after this one and create your three plots using matplotlib.

2.¶

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.

(a) Find an example of such a linear system whose graph is given by Figure 2


The plot of three intersecting planes in 3 dimensional space.
Figure 2

(b) Find an example of such a linear system whose graph is given by Figure 3


The plot of three intersecting planes in 3 dimensional space.
Figure 3

(c) Find an example of such a linear system whose graph is given by Figure 4


The plot of three intersecting planes in 3 dimensional space.
Figure 4

(d) Are there other graphs of a system of three linear equations with three unknown variables than the ones in this exercise? Create graphs of the other possible graphs of three equations with three unknowns in 3-dimensional space and include your graphs in your pdf with your responses.

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