Question

How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent `8x -9y = 27` and `9y -8x= -27`?

Answer 1

The equations are consistent but colinear, so there is no single solution. (see below for method).

Explanation:

For the equation `color(red)(8x-9y=27)`
one obvious solution pair can be obtained by setting `x=0` giving `(color(red)(0,-3))`
If we want a second pair with integer values, it will take a bit of playing but eventually we might come up with `(color(red)(9,5))`
These two pair of coordinates can be plotted on a Cartesian graph and connected with a straight line as in the first image below.

Similarly, for the equation `color(blue)(9y-8x=-27)`
we can obtain two pair of coordinates: `(color(blue)(0,-3))` and `(color(blue)(9,5))`.
Plotting these on a Cartesian graph and connecting with a straight line gives the second graph below.

Combining the two graphs results in overlapping, non-intersecting lines (third graph).
The graph are consistent in as much as they do not contradict each other, but they provide no single solution.