The Purplemath ForumsHelping students gain understanding and self-confidence in algebra powered by FreeFind

Systems of Non-Linear Equations:
Graphical Considerations
(page 2 of 6)

Suppose you have the following:

• Solve the system by graphing:
• y = x2
y
= 8 – x2

 I can graph each of these equations separately:   y = x2 y = 8 – x2

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

 Now look at the graph of the system: y = x2 y = 8 – x2

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).

So when you're trying to solve a system of equations, you're trying to find the coordinates of the intersection points.  Copyright © 2002-2011 Elizabeth Stapel All Rights Reserved

 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) = (13/7,  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.

<< Previous  Top  |  1 | 2 | 3 | 4 | 5 | 6  |  Return to Index  Next >>

 Cite this article as: Stapel, Elizabeth. "Systems of Non-Linear Equations: Graphical Considerations." Purplemath.     Available from http://www.purplemath.com/modules/syseqgen2.htm.     Accessed [Date] [Month] 2016

Purplemath:
Printing pages
School licensing

Reviews of
Internet Sites:
Free Help
Practice
Et Cetera

The "Homework
Guidelines"

Study Skills Survey

Tutoring from Purplemath
Find a local math tutor

This lesson may be printed out for your personal use.