Solving+System+of+Equations+Graphing

 **Solving Systems of Equations Graphing**  **__Quizzes__**  : http://www.quia.com/quiz/284838.html **__Notes:__** To solve a system of equation graphically, graph both equations and see where they intersect. The intersection point is the solution.



This picture is an example of a "consistent independent" graph. A consistent independent graph is a graph that has at least one solution. This is an example of a consistent dependent graph. In this graph you only have one line lying on the graph so there is only one possible answer.

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

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.

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

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. Below is an example an inconsistent system of equation

__** __**System of Equations**__-Two equations with two variables
 * __ Definitions:

**__intersect__**: meet (cross) at one point - this means there is **one solution** to the system of equation, and it is called the **simultaneous solution**. It is where the lines cross. **e.g. x + y = 4, 2x + y = 7** **__coincide:__** basically lie on top of each other - have all the same points that make it true - this means there are **infinitely many simultaneous solutions** to the system of equations. **e.g. x + y = 4, 2x + 2y = 8** **__parallel__**: the lines never meet/cross: (but are on the same graph) - this means there are **no simultaneous solutions** (no points are solutions of both equations). **e.g. x + y = 4, x + y = 5

__Consistent__**: A system of equation that has at least one solution.


 * __Inconsistent__**: A system of equation that has no soltion.

Ex. 1**
 * __Examples:__

y = 2x - 3 and y = x - 1 The graphs appear to intersect at the point with coordinates (2, 1). Check this estimate by replacing x with 2 and y with 1 in each equation. Check: The solution is (2, 1).
 * Ex. 2** Graph the system of equations to find the solution.
 * y = 2x - 3 || y = x - 1 ||
 * 1 = 2(2) - 3 || 1 = 2 - 1 ||
 * 1 = 1 || 1 = 1 ||

y = -x-3**
 * Ex. 3** Find the solution to the following system of equation
 * y = x+3

**__ Non Media Links: __**  [|**http://www.purplemath.com/modules/systlin2.htm**] http://www.e-tutor.com/et3/lessons/view/51184/print

__** http://www.youtube.com/watch?v=IN3JpyOSnRw http://www.youtube.com/watch?v=eD7FnWnCoQk&feature=related http://www.youtube.com/watch?v=lhvUYEYu8ko&feature=related __** Coffman, Joseph. __Solving System of Equations: Graphing__. [|http://http://jcoffman.com/]. __Graphing Systems of Equations__. <[|http://http://www.algebra-online.com/graphing-systems-equations-1.htm]>. __Solving Systems of Equations by Graphing__. .
 * __ (Youtube) Vidos:
 * __ Bibliography: