Most "solving by graphing" problems
work nicely, but sometimes they'll give you an inconsistent system (that
is, two parallel lines) or a dependent system (that is, two forms of the
same line equation). This is what these cases will look like:

With both equations solved for "y =", I can see that these two
equations are really both the same line! So the algebra tells me that this
is a dependent system, and the solution is the whole line. Of course,
this is a "solving by graphing" problem, so I still have to
do the graph, but I already know the answer.

solution: y = 36 – 9x

The solution to this system is the whole
line, so, in my classes, you could give the answer as being "y = 36 – 9x". However, most
books do something like this: You are looking for an x,y-solution,
and, in this case, x = x and y = 36 – 9x, so the solution "point"
is of the form (x,
36 – 9x). But then the book
does this weird thing with "a"
(or "t"
or "s"
or some other variable). Instead of using x,
which is a perfectly good variable, they pull out this new variable
from behind their left ear and give the solution as being "(a,
36 – 9a)".
I have no idea why they do this, but if your book does this, then (Warning!)
that is the format that your teacher will want on the test. Make sure
you memorize the variable that your particular book uses (which was "a"
in this example).

Solve the following
system by graphing.

7x + 2y = 16
–21x – 6y = 24

As usual, I'll first solve each equation
for "y =":

These lines have the same slope — namely, m = ^{–7}/_{2 } — but different y-intercepts,
so they are parallel. Since parallel lines never cross, the algebra
tells me that this is an inconsistent system; that is, there is no solution.
But this is a "solving by graphing" problem, so I still have
to draw the picture.

solution: no solution
(inconsistent system)

Warning: When the algebra tells you that
you have two parallel lines, for heaven's sake, draw the lines on your
graph so they look parallel!

Note: The solution to a dependent
system, being all the points along the line, contains infinitely-many
points. But don't make the mistake of thinking that "infinitely-many"
means "all". Any point off the line is not a solution;
only the infinity of points actually on the line will solve the
dependent system.

Also
note: The pictures on the
first page of this lesson are very useful for explaining "what's going
on" with linear systems, but pictures are not terribly useful
for finding actual solutions to systems. For instance, in the picture
at right, is the solution point at (–3,
2), or at (–3.15,
1.97)?

You can't tell!

Or, in the picture
at right, are the lines really parallel, so there's no solution?

Or are you just looking
at an un-useful portion of the graph?

You can't tell!

In this case, zooming
out shows that the lines in the previous picture do indeed cross,
at the point (450,
449.5). But this
was not at all apparent in the "standard" viewing window
shown above.

So you can see that the
pictures can be useful, especially for the concepts, but you should take
"solving by graphing" with a grain of salt, and should keep
in mind that the algebraic techniques (rather than mere pictures) are
the tools you need for solid answers.

The above discussion was
specific to the two-equation, two-variable case, because you can draw
pictures of the two-variable case to illustrate what is going on. But
the terminology and basic concepts are the same, no matter how many variables
you have. You could have four equations in four variables or twelve equations
in twelve variables, and you would still be looking for where the "lines" "intersect"
— you just couldn't draw a picture of it.

Formatting note: For reasons which will
become apparent when you start working with matrices,
equations in systems of equation are generally written with the variables
on the left-hand side of the "equals" sign and the numbers on
the right-hand side. Sometimes you'll find a question formatted differently,
but variables-on-the-left will be the norm.