AdvAlg5.3SolvingSystemsBySubstitution.notebook February 26, 2018

Transcription

1 Oct 14 4:53 PM 1

2 Why do we need algebraic techniques to solve systems? To find exact solutions The substitution method for solving systems of equations uses what property of equality? Substitution property of equality Oct 13 3:54 PM 2

3 Substitution Feb 20 12:46 PM 3

4 adult tickets (500) = 1000 student tickets (500) = 250 children's tickets Feb 20 12:47 PM 4

5 We have been working with systems of linear equations but the substitution method can also be used to solve what other type of system of equations? Non linear systems for example: Feb 20 12:46 PM 5

6 Substitution (3, 12) ( 3, 12) Feb 20 12:48 PM 6

7 What is a consistent system of equations? A system with one or more solutions What is an inconsistent system of equations? A system with no solutions How do we know if a system is inconsistent? The graphs do not intersect and Algebraically we get to an equation that is never true. i.e. 0 = 12 Oct 13 3:56 PM 7

8 These equations are both linear with the same slope and different y intercepts The graphs are parallel which results in no intersections and no solutions to the system of equations Feb 20 12:54 PM 8

9 3(2x 2 ) = 6x 2 Substitution 6x 2 = 6x 2 0=0 This result indicates that the system has many solutions The two equations are equivalent. The graphs coincide resulting in infinitely many solutions. We write this: All ordered pairs that satisfy y = 2x 2 Feb 20 12:55 PM 9

10 No intersections implies the system is inconsistent No solutions Two intersections implies the system is consistent with two solutions Same equation implies the system is consistent with infinite solutions Parallel lines, no intersections implies the system is inconsistent zero solutions Feb 20 12:55 PM 10

11 What indicates that a system has infinite solutions? The graphs are the same and Algebraically we get to an equation that is always true. i.e. 5 = 5 How do we write the answer to a system that has infinite solutions? The solution is: all ordered pairs that satisfy (then list any of the equations in the system) Oct 13 3:55 PM 11

12 Linear equations with different slopes. They will intersect resulting in one solution. 3x 8(2x + 9) = 7 Substitution 3x 16x 72 = 7 13x 72 = 7 13x = 65 x = 5 2( 5) + 9 = y 1 = y ( 5, 1) Feb 20 12:55 PM 12

13 Solve each. 8x + 8 ( 5 2x ) = 40 Sep 6 8:36 AM 13

14 Solve graphically. { x 2 2y = 4 2x + 5y = 25 Round all solutions to the hundredths Nov 19 7:04 AM 14

15 Why use Substitution? Algebraic method to get exact answer Consistent System: A system which has one or more solutions. 1. Aug 5 8:41 AM 15

16 2. 3. Aug 5 8:49 AM 16

17 4. A three bean salad can be made by mixing green, kidney and wax beans. The recipe calls for the same amount of kidney beans and wax beans and twice as much green beans as kidney beans. Determine how much of each kind of bean should be used for nine cups of salad. FIRST: Define your variables G = # of cups of green beans K = # of cups of kidney beans W = # of cups of wax beans Second: Develop your system from the given information. Write all mathematical sentences that present themselves.(this is the most important part of advanced mathematics) Aug 5 8:49 AM 17

18 5. One night at the circus the big top attraction sold out, selling all 3050 tickets. There are four times as many lower level seats as upper level seats. How many of each kind of seats are there? FIRST: Define your variables L = # of Lower level seats U = # of Upper level seats Second: Develop your system from the given information. Write all mathematical sentences that present themselves.(this is the most important part of advanced mathematics) Aug 5 8:53 AM 18

19 Inconsistent System: A system which has NO solutions. Aug 5 8:54 AM 19

20 { Determine whether the system is consistent or inconsistent. If it is consistent, tell how many solutions the system has. y = 12 / x 2 y = 2 x 2x y = 3 { y = 0 y = 3 / x 4x 2y = 6 { { x 2y = 4 2x 4y = 6 Sep 6 8:39 AM 20

21 Solve the system. 2 x y = 9 { 3 x 8 y = 7 Sep 6 8:44 AM 21

22 WS Lesson Masters (A).pdf Nov 12 9:49 AM 22

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31 WS Lesson Masters (A) KEY.pdf Nov 12 9:50 AM 31

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50 Attachments WS Lesson Masters A.pdf WS Lesson Masters A KEY.pdf