x y 3 amp y x 3 3 x 3 y 9 amp y x 3 Thats not a mistake the equations are

X y 3 amp y x 3 3 x 3 y 9 amp y x 3 thats not a

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x + y = 3 amp ; y = - x + 3 3 x + 3 y = 9 amp ; y = - x + 3 That’s not a mistake; the equations are identical. What does this mean about their graphs? They lie right on top of each 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 called coincident . A system of coincident lines is consistent has an infinite number of solutions. Such a system is called dependent (or consistent-dependent for clarity) In practice, consistent-dependent systems appear when you are given two pieces of equivalent information. Example D Returning to our xylophone and yam store, the yam salesman may say "We sold 4 times more yams than xylophones today!" while the xylophone salesman may say "We only sold one-fourth the number of xylophones as we did yams today." We can see that these two statements are saying the same thing in two different ways, and they will produce equivalent equations: y = 4 x amp ; y = 4 x x = 1 4 y amp ; y = 4 x Example E Determine if the system is consistent, inconsistent, or consistent-dependent 3 x - 2 y = 4 9 x - 6 y = 1 231
4.2. Types of Linear Systems www. c k12 .org Fist, put the equations into slope-intercept form: 3 x - 2 y = 4 amp ; y = 3 2 x - 2 9 x - 6 y = 1 amp ; y = 3 2 x - 1 6 We 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. Summary There 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 dependent Example F Two movie rental websites are in competition. Blamazon charges an annual membership of \$60 and charges \$3 per movie rental, while Nitflex charges an annual membership of \$40 and charges \$3 per movie rental. After how many movie 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 more expensive and it charges the same amount per movie as Nitflex. Let’s see how this works algebraically. Define the variables: Let x = number of movies rented and y = total rental cost y = 60 + 3 x Blamazon y = 40 + 3 x Nitflex The lines that describe each option have different y - intercepts, namely 60 for Blamazon and 40 for Nitflex. They have 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. MEDIA Click image to the left or use the URL below. URL: 232
www. c k12 .org Chapter 4. Unit 5 - Systems of Equations MEDIA Click image to the left or use the URL below.