Math 060 WORKSHEET

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A2T HW 1 - Solving Systems of Linear Equations by Graphing A system of linear equations looks like:  3

x

3

x

  2 4

y y

  7 10 NAME:_________________________ Graphically, each equation represents a line in the plane. When we consider two lines in the plane together, we ask certain questions: 1.) Do they intersect? 2.) If they do intersect, what is the point of intersection? 3.) Is it true that two arbitrary lines must intersect at a common point? 4.) Is it true that two arbitrary lines may intersect at more than one point? In this section we will answer these questions. A

solution

to a system of linear equations is an ordered pair that satisfies each equation in the system. On the graph, to be a solution to one equation the pair must lie on the line. To be a solution to a system means the pair must be on both lines simultaneously, a point of intersection. This tells us that we can solve a system by graphing each line and then determine the point of intersection by using the picture. Since this could be a guess, it is necessary to check the answer. EXAMPLES: Solve the following systems by graphing. Then check your answer. 1.) 2

x

 3

y

 6 2.)

x

 3

y

 11 3.)

y

 3

x

 1 4

x

 3

y

 12

x

 5

y

  13 2

y

 6

x

 4 1

4.)

x

  2

x

 3  .

5

y

 4 5.)

y

  3

y

3

x

x

6.)  2

x

4

x

 

y

2 

y

4   8 Now that we have seen all the possibilities, we can answer the above questions: 1.) Do they intersect? Not always. In this case, we say there is no solution. 2.) If they do intersect, what is the point of intersection? When we can find a single point of intersection, we write out the ordered pair as the solution. If there is more than one point of intersection, we say that there are infinitely many solutions. The system is called a

dependent

system. 3.) Is it true that two arbitrary lines must intersect at a common point? Not always. Parallel lines never intersect. This type of system is called

inconsistent

. We say that there is no solution. 4.) Is it true that two arbitrary lines may intersect at more than one point? Yes. Lines with the same slope and same

y

 intercept will intersect at every point on the line. This is how we get infinitely many solutions. This is a dependent system. Solve the following system by graphing:

x y

  1 9   2 2

y x

CONCLUSION: The graphing method does not work so well when the solution contains _________!

2

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