Question

How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent `3y − x = 4` and `y + 2x = −1`?

Answer 1

Graph each of the linear equations using a pair of points for each.
If the graphed lines intersect then the equations are consistent (and have a mutual solution at their intersection point).

Explanation:

For the equation `color(red)(3y-x=4)`
I found (arbitrary) solutions:
#color(white)("XXX")color(red){:(color(white)("X")x,color(white)("X")y),(-4,color(white)("X")0),(color(white)("X")2,color(white)("X")2) :}#

For the equation `color(blue)(y+2x=1)`
I found (again, arbitrary) solutions:
#color(white)("XXX")color(blue){:(color(white)("X")x,color(white)("X")y),(color(white)("X")0,color(white)("X")1),(color(white)("X")2,-3) :}#

Using each pair of points, we can plot lines:

or, combining them on a single graph:

We can see that the equations are consistent (since they intersect).