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Systems of Linear Equations: Graphing (page 2 of 7)

Sections: Definitions, Solving by graphing, Substitition, Elimination/addition, Gaussian elimination.


When you are solving systems, you are, graphically, finding intersections of lines. For two-variable systems, there are then three possible types of solutions:

Case 1

Case 2

Case 3

graph of intersecting lines

graph of parallel lines

graph with one apparent line

The first graph above, "Case 1", shows two distinct non-parallel lines that cross at exactly one point. This is called an "independent" system of equations, and the solution is always some x,y-point.

Independent system:
one solution point

Case 2

Case 3

graph of intersecting lines

graph of parallel lines

graph with one apparent line

The second graph above, "Case 2", shows two distinct lines that are parallel. Since parallel lines never cross, then there can be no intersection; that is, for a system of equations that graphs as parallel lines, there can be no solution. This is called an "inconsistent" system of equations, and it has no solution.

Independent system:
one solution and
one intersection point

Inconsistent system:
no solution and
no intersection point

Case 3

graph of intersecting lines

graph of parallel lines

graph with one apparent line

The third graph above, "Case 3", appears to show only one line. Actually, it's the same line drawn twice. These "two" lines, really being the same line, "intersect" at every point along their length. This is called a "dependent" system, and the "solution" is the whole line.

Independent system:
one solution and
one intersection point

Inconsistent system:
no solution and
no intersection point

Dependent system:
the solution is the
whole line

graph of intersecting lines

graph of parallel lines

graph with one apparent line

This shows that a system of equations may have one solution (a specific x,y-point), no solution at all, or an infinite solution (being all the solutions to the equation). You will never have a system with two or three solutions; it will always be one, none, or infinitely-many.


Probably the first method you'll see for solving systems of equations will be "solving by graphing". Warning: You have to take these problems with a grain of salt. The only way you can find the solution from the graph is IF you draw a very neat axis system, IF you draw very neat lines, IF the solution happens to be a point with nice neat whole-number coordinates, and IF the lines are not close to being parallel.  Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved

  

  

  
For instance, if the lines cross at a shallow angle it can be just about impossible to tell where the lines cross.

  

graph of lines with similar slopes
   

  

And if the intersection point isn't a neat pair of whole numbers, all bets are off.

  

  

(Can you tell by looking that the displayed solution has coordinates
of
(–4.3, –0.95)? No? Then you see my point.)

  

graph of lines with non-whole-number intersection point
   

On the plus side, since they will be forced to give you nice neat solutions for "solving by graphing" problems, you will be able to get all the right answers as long as you graph very neatly. For instance:

  • Solve the following system by graphing.
  •  

    ADVERTISEMENT

     

      2x – 3y = –2
      4
      x +  y = 24

     I know I need a neat graph, so I'll grab my ruler and get started. First, I'll solve each equation for "y=", so I can graph easily:

      2x – 3y = –2
      2x + 2 = 3y
      (2/3)x + (2/3) = y

      4x + y = 24
      y = –4x + 24

    The second line will be easy to graph using just the slope and intercept, but I'll need a T-chart for the first line.

      x

      y = (2/3)x + (2/3)

      y = –4x + 24

      –4

      –8/3 + 2/3 = –6/3 = –2

      16 + 24 = 40

      –1

      –2/3 + 2/3 = 0

      4 + 24 = 28

      2

      4/3 + 2/3 = 6/3 = 2

      –8 + 24 = 16

      5

      10/3 + 2/3 = 12/3 = 4

      –20 + 24 = 4

      8

      16/3 + 2/3 = 18/3 = 6

      –32 + 24 = –8

    (Sometimes you'll notice the intersection right on the T-chart. Do you see the point that is in both equations above? Check the gray-shaded row above.)

      
    Now that I have some points, I'll grab my ruler and graph neatly, and look for the intersection:

      

      

      

       

        

    Even if I hadn't noticed the intersection point in the T-chart, I can certainly see it from the picture.

      

    graph showing lines crossing at (5, 4)
      

      solution:  (x, y) = (5, 4)

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Cite this article as:

Stapel, Elizabeth. "Systems of Linear Equations: Graphing." Purplemath. Available from
    http://www.purplemath.com/modules/systlin2.htm. Accessed
 

 

 

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