..and each point on
each graph is a solution to that graph's equation.

Now look at the
graph of the system:

y
= x^{2}

y
= 8 – x^{2}

^{ }

^{ }

A solution to the system
is any point that is a solution for both equations. In other
words, a solution point for this system is any point that is on both
graphs. In other words:

"SOLUTIONS"
FOR SYSTEMS ARE INTERSECTIONS OF THE LINES

Then, graphically,
the solutions for this system are the red-highlighted points at
right:

That is, the solutions
to this system are the points (–2,
4) and (2,
4).

The system shown
above has two solutions, because the graph shows two intersection
points. A system can have one solution:

...lots of solutions:

...or no solutions
at all:

(In this last situation,
where there was no solution, the system of equations is said to be "inconsistent".)

When you look at
a graph, you can only guess at an approximation to the solution.
Unless the solutions points are nice neat numbers (and unless you
happen to know this in advance), you can't get the solution from
the picture. For instance, you can't tell what the solution
to the system graphed at right might be:

...because you're having
to guess from a picture. As it happens, the solution is (x,
y) = (1^{3}/_{7},^{ 9}/_{14}),
but you would have no possible way of knowing that from this picture.

Advisory: Your text will
almost certainly have you do some "solve by graphing" exercises. You
may safely assume for these exercises that answers are nice and neat,
because the solutions must be if you are to be able to have a chance
at guessing the solutions from a picture.

This "solving by graphing"
can be useful, in that it helps you get an idea in picture form of what
is going on when solving systems. But it can be misleading, too, in that
it implies that all solutions will be "neat" ones, when most
solutions are actually rather messy.

Stapel, Elizabeth.
"Systems of Non-Linear Equations: Graphical Considerations."
Purplemath. Available from http://www.purplemath.com/modules/syseqgen2.htm.
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