Chapter 4 - Systems of Linear Equations

Chapter 4 - Systems of Linear Equations

In this chapter we will consider situations in which the rela-tionship between two variables is described by a pair of linearequations. Groups of equations are called systems of equationsand they are used in a wide variety of business and science ap-plications.

One goal in our work with systems of equations is to find asolution to the system, if it exists. We will have several di↵er-ent methods by the end of this chapter, including a graphicalapproach. Through graphing you will first see what types ofsolution sets to expect and you will also quickly see how we willneed more e�cient methods of finding solutions!

Example #1: Graph the following system (i.e.: graph bothlines together):

x� y = 2x+ y = 4

At what point do the lines intersect? Use your graph to findthe point.

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Example #2: Graph the following system:

x+ 5y = 6x+ 5y = 0

Do the lines intersect? Again, use your graph to answer thisquestion.

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Example #3: Graph the following system:

x = 5 + y�2x+ 2y = �10

Do the lines intersect?

These lines should intersect infinitely often because they areactually the same line. In this case, we say the system hasinfinitely many solutions.

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We have just seen three di↵erent types of systems of equations.In order to highlight the di↵erences between the three exampleswe just looked at let’s answer the following questions:

1. How many solutions (i.e.: points) did you find in Ex. #1?

2. How many solutions did you find in Example #2?

3. How many solutions did you find in Example #3?

4. What made Example #2 di↵erent from the otherexamples? Briefly explain why you could not find asolution in this case.

5. Can two lines intersect in two points? three points? npoints? Use graphs to help you answer this question.

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Now let’s apply what you’ve learned to a practical example.Here’s a business scenario where we can use a system ofequations to find out the break-even point (where your incomeequals your expenses) in the printing industry.

Example #4:

To print a newsletter costs $450 fixed overhead plus $1.50 a copy.Every printed copy of newsletter is sold at $3 a piece.

1. If x represents the number of copies printed and y is theamount of money it costs to print the newsletter, write anequation that relates x and y.

2. If x represents the number of copies printed and y is theamount of income from newsletter sales, write an equationthat relates x and y.

3. Graph the two lines you found above on the samecoordinate plane.

4. Find the break-even point (the point of intersection) forprinting the newsletter.

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