
Systems of Linear Equations: Graphing (page 2 of 7) Sections: Definitions, Solving by graphing, Substitition, Elimination/addition, Gaussian elimination. When you are solving systems of equations (linear or otherwise), you are, in terms of the equations' related graphed lines, finding any intersection points of those lines. For twovariable linear systems of equations, there are then three possible types of solutions to the systems, which correspond to three different types of graphs of two straight lines. These three cases are illustrated below:
The first graph above, "Case 1", shows two distinct nonparallel lines that cross at exactly one point. This is called an "independent" system of equations, and the solution is always some x,ypoint.
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.
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.
This shows that a system of equations may have one solution (a specific x,ypoint), 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 infinitelymany. 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 wholenumber coordinates, and IF the lines are not close to being parallel. Copyright © Elizabeth Stapel 20032011 All Rights Reserved
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:
2x – 3y = –2 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 4x + y = 24 The second line will be easy to graph using just the slope and intercept, but I'll need a Tchart for the first line.
(Sometimes you'll notice the intersection right on the Tchart. Do you see the point that is in both equations above? Check the grayshaded row above.)
solution: (x, y) = (5, 4) << Previous Top  1  2  3  4  5  6  7  Return to Index Next >>


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