Determining the Best Method to Solve a Linear System

Transcription

1 Determining the Best Method to Solve a Linear System Lori Jordan Kate Dirga Say Thanks to the Authors Click (No sign in required)

2 To access a customizable version of this book, as well as other interactive content, visit AUTHORS Lori Jordan Kate Dirga CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook textbooks). Copyright 2015 CK-12 Foundation, The names CK-12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License ( licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at terms-of-use. Printed: April 5, 2015

3 Chapter 1. Determining the Best Method to Solve a Linear System CHAPTER 1 Determining the Best Method to Solve a Linear System Here you will solve systems of linear equations using the most efficient method. A rental car company, Affordable Autos, charges $30 per day plus $0.51 per mile driven. A second car rental company, Cheap Cars, charges $25 per day plus $0.57 per mile driven. For a short distance, Cheap Cars offers the better deal. At what point (after how many miles in a single day) does the Affordable Autos rental company offer the better deal? Guidance Any of the methods (graphing, substitution, linear combination) learned in this unit can be used to solve a linear system of equations. Sometimes, however it is more efficient to use one method over another based on how the equations are presented. For example If both equations are presented in slope intercept form (y = mx+b), then either graphing or substitution would be most efficient. If one equation is given in slope intercept form or solved for x, then substitution might be easiest. If both equations are given in standard form (Ax +By = C), then linear combinations is usually most efficient. Example A Solve the following system: y = x + 5 y = 1 2 x + 2 Solution: Since both equations are in slope intercept form we could easily graph these lines. The question is whether or not the intersection of the two lines will lie on the grid (whole numbers). If not, it is very difficult to determine an answer from a graph. One way to get around this difficulty is to use technology to graph the lines and find their intersection. The first equation has a y intercept of 5 and slope of -1. It is shown here graphed in blue. The second equation has a y intercept of 2 and a slope of 1 2. It is shown here graphed in red. The two lines clearly intersect at (2, 3). 1

4 Alternate Method: Substitution may be the preferred method for students who would rather solve equations algebraically. Since both of these equations are equal to y, we can let the right hand sides be equal to each other and solve for x: x + 5 = 1 2 x ( x + 5 = 12 ) x + 2 Multiplying the equation by 2 eliminates the fraction. 2x + 10 = x = 3x x = 2 Now solve for y: y = (2) + 5 y = 3 Solution: (2, 3) Check your answer: 3 = = Example B Solve the system: 2

5 Chapter 1. Determining the Best Method to Solve a Linear System 15x + y = 24 y = 4x + 2 Solution: This time one of our equations is already solved for y. It is easiest here to use this expression to substitute into the other equation and solve: 15x + ( 4x + 2) = 24 15x 4x + 2 = 24 11x = 22 x = 2 Now solve for y: y = 4(2) + 2 y = y = 6 Solution: (2, -6) Check your answer: 15(2) + ( 6) = 30 6 = 24 6 = 4(2) + 2 = = 6 Example C Solve the system: 6x + 11y = 86 9x 13y = 115 Solution: Both equations in this example are in standard form so the easiest method to use here is linear combinations. Since the LCM of 6 and 9 is 18, we will multiply the first equation by 3 and the second equation by 2 to eliminate x first: 3( 6x + 11y = 86) 18x + 33y = 258 2(9x 13y = 115) 18x 26y = 230 7y = 28 y = 4 Now solve for x: 3

6 6x + 11(4) = 86 6x + 44 = 86 6x = 42 x = 7 Solution: (-7, 4) Check your answer: 6( 7) + 11(4) = = 86 9( 7) 13(4) = = 115 Intro Problem Revisit Set up equations to represent the total cost (for one day s rental) for each company: Affordable Autos y = 0.51x + 30 Cheap cars y = 0.57x + 25 It is easiest to use substitution here. Substituting y = 0.51x + 30 into the second equation, we get: 0.51x + 30 = 0.57x = 0.06x x = Therefore, Affordable Autos has a better deal if we want to drive more than 83 and one-third miles during our one-day rental. Guided Practice Solve the following systems using the most efficient method: 1. y = 3x + 2 y = 2x x + 5y = 5 x = 2y x 5y = 24 15x + 7y = 4 4

7 Chapter 1. Determining the Best Method to Solve a Linear System Answers 1. This one could be solved by graphing, graphing with technology or substitution. This time we will use substitution. Since both equations are solved for y, we can set them equal and solve for x: 3x + 2 = 2x 3 5 = 5x x = 1 Now solve for y: y = 3(1) + 2 y = y = 1 Solution: (1, -1) 2. Since the second equation here is solved for x, it makes sense to use substitution: 4(2y 11) + 5y = 5 8y y = 5 13y = 39 y = 3 Now solve for x: x = 2(3) 11 x = 6 11 x = 5 Solution: (-5, 3) 3. This time, both equations are in standard form so it makes the most sense to use linear combinations. We can eliminate y by multiplying the first equation by 7 and the second equation by 5: 7(4x 5y = 24) 28x 35y = 168 5( 15x + 7y = 4) 75x + 35y = 20 47x = 188 x = 4 Now find y: 5

8 4(4) 5y = y = 24 5y = 40 y = 8 Solution: (4, 8) Explore More Solve the following systems using linear combinations x 2y = 1 8x + 4y = x + y = 16 4x y = x + 2y = 4 y = 4x x + 5y = 25 x = 2y x + 10y = 1 2x 6y = 2 6

9 Chapter 1. Determining the Best Method to Solve a Linear System 6.. 3x + y = 18 7x + 3y = x + 15y = 3 3x 5y = x y = 19 13x + 2y = x 15y = 6 x = 9y x 4y = 1 2x + 3y = x y = 2 3x 2y = x + 12y = 18 y = 1 4 x 3 2 7

10 x 8y = 2 x = 1 2 y x + y = 3 21x 3y = x 10y = 2 y = 4 5 x + 7 Solve the following word problem by creating and solving a system of linear equations. 16. Jack and James each buy some small fish for their new aquariums. Jack buys 10 clownfish and 7 goldfish for $ James buys 5 clownfish and 6 goldfish for $ How much does each type of fish cost? 17. The sum of two numbers is 35. The larger number is one less than three times the smaller number. What are the two numbers? 18. Rachel offers to go to the coffee shop to buy cappuccinos and lattes for her coworkers. She buys a total of nine drinks for $ If cappuccinos cost $3.75 each and the lattes cost $4.25 each, how many of each drink did she buy? 8