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x+y=3→amp;y=-x+33x+3y=9→amp;y=-x+3That’s not a mistake; the equations are identical. What does this mean about their graphs? They lie right on top ofeach other!How many times do these lines ’intersect’? An infinite number! Every point on the line is a point of intersection.Graphs that lie precisely on top of one another are calledcoincident. A system of coincident lines is consistent hasan infinite number of solutions. Such a system is calleddependent(orconsistent-dependentfor clarity)In practice, consistent-dependent systems appear when you are given two pieces of equivalent information.Example DReturning to our xylophone and yam store, the yam salesman may say "We sold 4 times more yams than xylophonestoday!" while the xylophone salesman may say "We only sold one-fourth the number of xylophones as we did yamstoday." We can see that these two statements are saying the same thing in two different ways, and they will produceequivalent equations:y=4x→amp;y=4xx=14y→amp;y=4xExample EDetermine if the system is consistent, inconsistent, or consistent-dependent3x-2y=49x-6y=1231
4.2. Types of Linear Systemswww.ck12.orgFist, put the equations into slope-intercept form:3x-2y=4→amp;y=32x-29x-6y=1→amp;y=32x-16We see that these lines have the same slope but different y-intercepts. Therefore:• These lines are parallel.• The system has no solution.• The system is inconsistent.SummaryThere are three possibilities for the solutions to a linear system:• One solution - Consistent and Independent• No solutions - Inconsistent• An infinite number of solutions - Consistent and dependentExample FTwo movie rental websites are in competition. Blamazon charges an annual membership of $60 and charges $3 permovie rental, while Nitflex charges an annual membership of $40 and charges $3 per movie rental. After how manymovie rentals would Blamazon become the better option?It should already be clear to see that Blamazon will never become the better option, since its membership is moreexpensive and it charges the same amount per movie as Nitflex.Let’s see how this works algebraically.Define the variables: Letx=number of movies rented andy=total rental costy=60+3xBlamazony=40+3xNitflexThe lines that describe each option have differenty-intercepts, namely 60 for Blamazon and 40 for Nitflex. Theyhave the same slope, three dollars per movie. This means that the lines are parallel and the system is inconsistent.The system has no solutions, so there is no number of rentals for which the two websites will have the same cost.MEDIAClick image to the left or use the URL below.URL:232
www.ck12.orgChapter 4.Unit 5 - Systems of EquationsMEDIAClick image to the left or use the URL below.