Atividade de Algebra 2 em Inglês 3 Atividades Separadas

NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S158 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 31 Systems of Equations Classwork Exploratory Challenge 1 a Sketch the lines given by 𝑥 𝑦 6 and 3𝑥 𝑦 2 on the same set of axes to solve the system graphically Then solve the system of equations algebraically to verify your graphical solution b Suppose the second line is replaced by the line with equation 𝑥 𝑦 2 Plot the two lines on the same set of axes and solve the pair of equations algebraically to verify your graphical solution NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S159 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License c Suppose the second line is replaced by the line with equation 2𝑥 12 2𝑦 Plot the lines on the same set of axes and solve the pair of equations algebraically to verify your graphical solution d We have seen that a pair of lines can intersect in 1 0 or an infinite number of points Are there any other possibilities NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S160 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exploratory Challenge 2 a Suppose that instead of equations for a pair of lines you were given an equation for a circle and an equation for a line What possibilities are there for the two figures to intersect Sketch a graph for each possibility b Graph the parabola with equation 𝑦 𝑥2 What possibilities are there for a line to intersect the parabola Sketch each possibility NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S161 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License c Sketch the circle given by 𝑥2 𝑦2 1 and the line given by 𝑦 2𝑥 2 on the same set of axes One solution to the pair of equations is easily identifiable from the sketch What is it d Substitute 𝑦 2𝑥 2 into the equation 𝑥2 𝑦2 1 and solve the resulting equation for 𝑥 e What does your answer to part d tell you about the intersections of the circle and the line from part c Exercises 1 Draw a graph of the circle with equation 𝑥2 𝑦2 9 a What are the solutions to the system of circle and line when the circle is given by 𝑥2 𝑦2 9 and the line is given by 𝑦 2 NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S162 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License b What happens when the line is given by 𝑦 3 c What happens when the line is given by 𝑦 4 2 By solving the equations as a system find the points common to the line with equation 𝑥 𝑦 6 and the circle with equation 𝑥2 𝑦2 26 Graph the line and the circle to show those points NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S163 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Graph the line given by 5𝑥 6𝑦 12 and the circle given by 𝑥2 𝑦2 1 Find all solutions to the system of equations 4 Graph the line given by 3𝑥 4𝑦 25 and the circle given by 𝑥2 𝑦2 25 Find all solutions to the system of equations Verify your result both algebraically and graphically 5 Graph the line given by 2𝑥 𝑦 1 and the circle given by 𝑥2 𝑦2 10 Find all solutions to the system of equations Verify your result both algebraically and graphically NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S164 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 6 Graph the line given by 𝑥 𝑦 2 and the quadratic curve given by 𝑦 𝑥2 4 Find all solutions to the system of equations Verify your result both algebraically and graphically NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S165 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Where do the lines given by 𝑦 𝑥 𝑏 and 𝑦 2𝑥 1 intersect 2 Find all solutions to the following system of equations 𝑥 22 𝑦 32 4 𝑥 𝑦 3 Illustrate with a graph 3 Find all solutions to the following system of equations 𝑥 2𝑦 0 𝑥2 2𝑥 𝑦2 2𝑦 3 0 Illustrate with a graph 4 Find all solutions to the following system of equations 𝑥 𝑦 4 𝑥 32 𝑦 22 10 Illustrate with a graph Lesson Summary Here are some steps to consider when solving systems of equations that represent a line and a quadratic curve 1 Solve the linear equation for 𝑦 in terms of 𝑥 This is equivalent to rewriting the equation in slopeintercept form Note that working with the quadratic equation first would likely be more difficult and might cause the loss of a solution 2 Replace 𝑦 in the quadratic equation with the expression involving 𝑥 from the slopeintercept form of the linear equation That will yield an equation in one variable 3 Solve the quadratic equation for 𝑥 4 Substitute 𝑥 into the linear equation to find the corresponding value of 𝑦 5 Sketch a graph of the system to check your solution NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 31 ALGEBRA II Lesson 31 Systems of Equations S166 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from ALG IIM1TE130072015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 5 Find all solutions to the following system of equations 𝑦 2𝑥 3 𝑦 𝑥2 6𝑥 3 Illustrate with a graph 6 Find all solutions to the following system of equations 𝑦2 6𝑦 𝑥 9 0 6𝑦 𝑥 27 Illustrate with a graph 7 Find all values of 𝑘 so that the following system has two solutions 𝑥2 𝑦2 25 𝑦 𝑘 Illustrate with a graph 8 Find all values of 𝑘 so that the following system has exactly one solution 𝑦 5 𝑥 32 𝑦 𝑘 Illustrate with a graph 9 Find all values of 𝑘 so that the following system has no solutions 𝑥2 𝑦 𝑘2 36 𝑦 5𝑥 𝑘 Illustrate with a graph NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 PRECALCULUS AND ADVANCED TOPICS Lesson 16 Solving General Systems of Linear Equations S112 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 16 Solving General Systems of Linear Equations Classwork Example A scientist measured the greatest linear dimension of several irregular metal objects He then used water displacement to calculate the volume of each of the objects The data he collected are 1 3 2 5 4 9 and 6 20 where the first coordinate represents the linear measurement of the object in centimeters and the second coordinate represents the volume in cubic centimeters Knowing that volume measures generally vary directly with the cubed value of linear measurements he wants to try to fit this data to a curve in the form of 𝑣𝑥 𝑎𝑥3 𝑏𝑥2 𝑐𝑥 𝑑 a Represent the data using a system of equations b Represent the system using a matrix equation in the form 𝐴𝑥 𝑏 c Use technology to solve the system d Based on your solution to the system what cubic equation models the data e What are some of the limitations of the model NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 PRECALCULUS AND ADVANCED TOPICS Lesson 16 Solving General Systems of Linear Equations S113 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercises 1 An attendance officer in a small school district noticed a trend among the four elementary schools in the district This district used an open enrollment policy which means any student within the district could enroll at any school in the district Each year 10 of the students from Adams Elementary enrolled at Davis Elementary and 10 of the students from Davis enrolled at Adams In addition 10 of the students from Brown Elementary enrolled at Carson Elementary and 20 of the students from Brown enrolled at Davis At Carson Elementary about 10 of students enrolled at Brown and 10 enrolled at Davis while at Davis 10 enrolled at Brown and 20 enrolled at Carson The officer noted that this year the enrollment was 490 250 300 and 370 at Adams Brown Carson and Davis respectively a Represent the relationship that reflects the annual movement of students among the elementary schools using a matrix b Write an expression that could be used to calculate the attendance one year prior to the year cited by the attendance officer Find the enrollment for that year c Assuming that the trend in attendance continues write an expression that could be used to calculate the enrollment two years after the year cited by the attendance officer Find the attendance for that year d Interpret the results to part c in context NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 PRECALCULUS AND ADVANCED TOPICS Lesson 16 Solving General Systems of Linear Equations S114 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 2 Mrs Kenrick is teaching her class about different types of polynomials They have just studied quartics and she has offered 5 bonus points to anyone in the class who can determine the quartic that she has displayed on the board The quartic has 5 points identified 6 25 3 1 2 7 3 0 5 and 3 169 Logan really needs those bonus points and remembers that the general form for a quartic is 𝑦 𝑎𝑥4 𝑏𝑥3 𝑐𝑥2 𝑑𝑥 𝑒 Can you help Logan determine the equation of the quartic a Write the system of equations that would represent this quartic b Write a matrix that would represent the coefficients of this quartic c Write an expression that could be used to calculate coefficients of the equation d Explain the answer in the context of this problem NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 PRECALCULUS AND ADVANCED TOPICS Lesson 16 Solving General Systems of Linear Equations S115 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 The Fibonacci numbers are the numbers 1 1 2 3 5 8 13 21 34 Each number beyond the second is the sum of the previous two Let 𝑢1 1 1 𝑢2 1 2 𝑢3 2 3 𝑢4 3 5 𝑢5 5 8 and so on a Show that 𝑢𝑛1 0 1 1 1 𝑢𝑛 b How could you use matrices to find 𝑢30 Use technology to find 𝑢30 c If 𝑢𝑛 165580141 267914296 find 𝑢𝑛1 Show your work NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 PRECALCULUS AND ADVANCED TOPICS Lesson 16 Solving General Systems of Linear Equations S116 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 The system of equations is given 12𝑥 3𝑦 5𝑧 42𝑤 𝑣 0 6𝑥 5𝑦 2𝑤 3𝑦 45𝑧 6𝑤 2𝑣 10 9𝑥 𝑦 𝑧 2𝑣 3 4𝑥 2𝑦 𝑤 3𝑣 1 a Represent this system using a matrix equation b Use technology to solve the system Show your solution process and round your entries to the tenths place 2 A caterer is preparing a fruit salad for a party She decides to use strawberries blackberries grapes bananas and kiwi The total weight of the fruit is 10 pounds Based on guidelines from a recipe the weight of the grapes is equal to the sum of the weight of the strawberries and blackberries the total weight of the blackberries and kiwi is 2 pounds half the total weight of fruit consists of kiwi strawberries and bananas and the weight of the grapes is twice the weight of the blackberries a Write a system of equations to represent the constraints placed on the caterer when she makes the fruit salad Be sure to define your variables b Represent the system using a matrix equation c Solve the system using the matrix equation Explain your solution in context d How helpful would the solution to this problem likely be to the caterer as she prepares to buy the fruit 3 Consider the sequence 1 1 1 3 5 9 17 31 57 where each number beyond the third is the sum of the previous three Let 𝑤𝑛 be the points with the 𝑛th 𝑛 1th and 𝑛 2th terms of the sequence a Find a 3 3 matrix 𝐴 so that 𝐴𝑤𝑛 𝑤𝑛1 for each 𝑛 b What is the 30th term of the sequence c What is 𝐴1 Explain what 𝐴1 represents in terms of the sequence In other words how can you find 𝑤𝑛1 if you know 𝑤𝑛 d Could you find the 5th term in the sequence If so how What is its value 4 Mr Johnson completes a survey on the number of hours he spends weekly watching different types of television programs He determines that he spends 30 hours a week watching programs of the following types comedy drama movies competition and sports He spends half as much time watching competition shows as he does watching dramas His time watching sports is double his time watching dramas He spends an equal amount of time watching comedies and movies The total amount of time he spends watching comedies and movies is the same as the total amount of time he spends watching dramas and competition shows Write and solve a system of equations to determine how many hours Mr Johnson watches each type of programming each week NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 PRECALCULUS AND ADVANCED TOPICS Lesson 16 Solving General Systems of Linear Equations S117 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 5 A copper alloy is a mixture of metals having copper as their main component Copper alloys do not corrode easily and conduct heat They are used in all types of applications including cookware and pipes A scientist is studying different types of copper alloys and has found one containing copper zinc tin aluminum nickel and silicon The alloy weighs 32 kilograms The percentage of aluminum is triple the percentage of zinc The percentage of silicon is half that of zinc The percentage of zinc is triple that of nickel The percentage of copper is fifteen times the sum of the percentages of aluminum and zinc combined The percentage of copper is nine times the combined percentages of all the other metals a Write and solve a system of equations to determine the percentage of each metal in the alloy b How many kilograms of each alloy are present in the sample NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 PRECALCULUS AND ADVANCED TOPICS Lesson 2 Networks and Matrix Arithmetic S12 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 2 Networks and Matrix Arithmetic Classwork Opening Exercise Suppose a subway line also connects the four cities Here is the subway and bus line network The bus routes connecting the cities are represented by solid lines and the subway routes are represented by dashed arcs Write a matrix 𝐵 to represent the bus routes and a matrix 𝑆 to represent the subway lines connecting the four cities Exploratory ChallengeExercises 16 Matrix Arithmetic Use the network diagram from the Opening Exercise and your answers to help you complete this challenge with your group 1 Suppose the number of bus routes between each city were doubled a What would the new bus route matrix be NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 PRECALCULUS AND ADVANCED TOPICS Lesson 2 Networks and Matrix Arithmetic S13 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License b Mathematicians call this matrix 2𝐵 Why do you think they call it that 2 What would be the meaning of 10𝐵 in this situation 3 Write the matrix 10𝐵 4 Ignore whether or not a line connecting cities represents a bus or subway route a Create one matrix that represents all the routes between the cities in this transportation network b Why would it be appropriate to call this matrix 𝐵 𝑆 Explain your reasoning NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 PRECALCULUS AND ADVANCED TOPICS Lesson 2 Networks and Matrix Arithmetic S14 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 5 What would be the meaning of 4𝐵 2𝑆 in this situation 6 Write the matrix 4𝐵 2𝑆 Show work and explain how you found your answer NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 PRECALCULUS AND ADVANCED TOPICS Lesson 2 Networks and Matrix Arithmetic S15 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 7 7 Complete this graphic organizer Matrix Operations Graphic Organizer Operation Symbols Describe How to Calculate Example Using 𝟑 𝟑 Matrices Scalar Multiplication 𝑘𝐴 The Sum of Two Matrices 𝐴 𝐵 The Difference of Two Matrices 𝐴 𝐵 𝐴 1𝐵 NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 PRECALCULUS AND ADVANCED TOPICS Lesson 2 Networks and Matrix Arithmetic S16 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 For the matrices given below perform each of the following calculations or explain why the calculation is not possible 𝐴 1 2 0 1 𝐵 2 1 1 4 𝐶 5 2 9 6 1 3 1 1 0 𝐷 1 6 0 3 0 2 1 3 2 a 𝐴 𝐵 b 2𝐴 𝐵 c 𝐴 𝐶 d 2𝐶 e 4𝐷 2𝐶 f 3𝐵 3𝐵 g 5𝐵 𝐶 h 𝐵 3𝐴 i 𝐶 10𝐷 j 1 2 𝐶 𝐷 k 1 4 𝐵 l 3𝐷 4𝐴 m 1 3 𝐵 2 3 𝐴 Lesson Summary MATRIX SCALAR MULTIPLICATION Let 𝑘 be a real number and let 𝐴 be an 𝑚 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖𝑗 Then the scalar product 𝑘 𝐴 is the 𝑚 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑘 𝑎𝑖𝑗 MATRIX SUM Let 𝐴 be an 𝑚 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖𝑗 and let 𝐵 be an 𝑚 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑏𝑖𝑗 Then the matrix sum 𝐴 𝐵 is the 𝑚 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖𝑗 𝑏𝑖𝑗 MATRIX DIFFERENCE Let 𝐴 be an 𝑚 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖𝑗 and let 𝐵 be an 𝑚 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑏𝑖𝑗 Then the matrix difference 𝐴 𝐵 is the 𝑚 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖𝑗 𝑏𝑖𝑗 NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 PRECALCULUS AND ADVANCED TOPICS Lesson 2 Networks and Matrix Arithmetic S17 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 2 For the matrices given below perform each of the following calculations or explain why the calculation is not possible 𝐴 1 2 1 3 0 2 𝐵 2 1 3 6 1 0 𝐶 1 2 3 1 1 4 𝐷 2 1 1 0 4 1 a 𝐴 2𝐵 b 2𝐴 𝐶 c 𝐴 𝐶 d 2𝐶 e 4𝐷 2𝐶 f 3𝐷 3𝐷 g 5𝐵 𝐷 h 𝐶 3𝐴 i 𝐵 10𝐷 j 1 2 𝐶 𝐴 k 1 4 𝐵 l 3𝐴 3𝐵 m 1 3 𝐵 2 3 𝐷 3 Let 𝐴 3 2 3 1 5 and 𝐵 1 2 3 2 4 1 a Let 𝐶 6𝐴 6𝐵 Find matrix 𝐶 b Let 𝐷 6𝐴 𝐵 Find matrix 𝐷 c What is the relationship between matrices 𝐶 and 𝐷 Why do you think that is 4 Let 𝐴 3 2 1 5 3 4 and 𝑋 be a 3 2 matrix If 𝐴 𝑋 2 3 4 1 1 5 then find 𝑋 NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 PRECALCULUS AND ADVANCED TOPICS Lesson 2 Networks and Matrix Arithmetic S18 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 5 Let 𝐴 1 3 2 3 1 2 4 3 2 and 𝐵 2 1 3 2 2 1 1 3 1 represent the bus routes of two companies between three cities a Let 𝐶 𝐴 𝐵 Find matrix 𝐶 Explain what the resulting matrix and entry 𝑐13 mean in this context b Let 𝐷 𝐵 𝐴 Find matrix 𝐷 Explain what the resulting matrix and entry 𝑑13 mean in this context c What is the relationship between matrices 𝐶 and 𝐷 Why do you think that is 6 Suppose that Aprils Pet Supply has three stores in cities 1 2 and 3 Bens Pet Mart has two stores in cities 1 and 2 Each shop sells the same type of dog crates in sizes 1 small 2 medium 3 large and 4 extra large Aprils and Bens inventory in each city is stored in the tables below Aprils Pet Supply Bens Pet Mart City 1 City 2 City 3 City 1 City 2 Size 1 3 5 1 Size 1 2 3 Size 2 4 2 9 Size 2 0 2 Size 3 1 4 2 Size 3 4 1 Size 4 0 0 1 Size 4 0 0 a Create a matrix 𝐴 so that 𝑎𝑖𝑗 represents the number of crates of size 𝑖 available in Aprils store in City 𝑗 b Explain how the matrix 𝐵 2 3 0 0 2 0 4 1 0 0 0 0 can represent the dog crate inventory at Bens Pet Mart c Suppose that April and Ben merge their inventories Find a matrix that represents their combined inventory in each of the three cities NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 PRECALCULUS AND ADVANCED TOPICS Lesson 2 Networks and Matrix Arithmetic S19 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 7 Jackie has two businesses she is considering buying and a business plan that could work for both Consider the tables below and answer the questions following Horuss OneStop Warehouse Supply Res 24Hour Distributions If Business Stays the Same If Business Improves as Projected If Business Stays the Same If Business Improves as Projected Expand to Multiple States 75000000 45000000 Expand to Multiple States 99000000 62500000 Invest in Drone Delivery 33000000 30000000 Invest in Drone Delivery 49000000 29000000 Close and Sell Out 20000000 20000000 Close and Sell Out 35000000 35000000 a Create matrices 𝐻 and 𝑅 representing the values in the tables above such that the rows represent the different options and the columns represent the different outcomes of each option b Calculate 𝑅 𝐻 What does 𝑅 𝐻 represent c Calculate 𝐻 𝑅 What does 𝐻 𝑅 represent d Jackie estimates that the economy could cause fluctuations in her numbers by as much as 5 both ways Find matrices to represent the best and worst case scenarios for Jackie e Which business should Jackie buy Which of the three options should she choose Explain your choices For 13 𝑎1 3 𝑏1 2 𝑐1 𝑑 3 𝑎 𝑏 𝑐 𝑑 3 For 25 𝑎2 3 𝑏2 2 𝑐2 𝑑 5 8𝑎 4𝑏 2𝑐 𝑑 5 For 49 𝑎4 3 𝑏4 2 𝑐4 𝑑 9 64𝑎 16𝑏 4𝑐 𝑑 9 For 620 𝑎6 3 𝑏6 2 𝑐6 𝑑 20 216𝑎 36𝑏 6𝑐 𝑑 20 System of equations 𝑎 𝑏 𝑐 𝑑 3 8𝑎 4𝑏 2𝑐 𝑑 5 64𝑎 16𝑏 4𝑐 𝑑 9 216𝑎 36𝑏 6𝑐 𝑑 20 A is the matrix of coefficients x is the vector 𝑎 𝑏 𝑐 𝑑 𝑇 b is the vector of constants on the righthand side Once we have solved for aaa bbb ccc and ddd we substitute those values into the cubic equation 𝑣𝑥 𝑎𝑥 3 𝑏𝑥 2 𝑐𝑥 𝑑 The model assumes a cubic relationship between the linear dimensions and volume which may not always perfectly reflect reality The data points are limited in number so the model may not generalize well to other objects The measurements themselves may contain errors impacting the accuracy of the model 𝑥𝑝𝑟𝑖𝑜𝑟 𝑀 1𝑥 This requires finding the inverse of matrix M which can be computed using standard matrix inversion techniques Once is found multiplying it by the current enrollment vector will 𝑀 1 𝑥 give the enrollment from one year ago 𝑥2𝑦𝑒𝑎𝑟𝑠 𝑀 2𝑥 Here represents the transition matrix after two years 𝑀 2 The results show how the distribution of students changes over time based on the transition matrix If the transition percentages remain constant the student population across the schools will reach a steady state where the movement of students between schools stabilizes For 625 𝑎 6 4 𝑏 6 3 𝑐 6 2 𝑑 6 𝑒 25 For 31 𝑎 3 4 𝑏 3 3 𝑐 3 2 𝑑 3 𝑒 1 For 2 3 0 𝑎 2 3 4 𝑏 2 3 3 𝑐 2 3 2 𝑑 2 3 𝑒 0 For 05 𝑒 5 For 3169 𝑎3 4 𝑏3 3 𝑐3 2 𝑑3 𝑒 169 System of equations 𝑎 6 4 𝑏 6 3 𝑐 6 2 𝑑 6 𝑒 25 𝑎 3 4 𝑏 3 3 𝑐 3 2 𝑑 3 𝑒 1 𝑎 2 3 4 𝑏 2 3 3 𝑐 2 3 2 𝑑 2 3 𝑒 0 𝑒 5 We can solve for the coefficients a b c and d by solving the matrix equation 𝐴𝑥 𝑏 This could be done by either performing Gaussian elimination or by finding the inverse of matrix A if it is invertible In this case the matrix A represents the relationship between the powers of 6 3 and 2 3 for the variables a b c and d Solving the matrix will give us the values for these coefficients which when plugged back into the quartic equation will satisfy the conditions given by the system Given that Fibonacci numbers follow the rule where each number is the sum of the two preceding ones we define the sequence in matrix form and So this shows the matrix multiplication gives the correct Fibonacci sequence relationship 𝐹𝑛 165580141 𝐹𝑛1 267914296 𝐹𝑛2 267914296 165580141 102334155 s weight of strawberries in pounds b weight of blackberries in pounds g weight of grapes in pounds n weight of bananas in pounds k weight of kiwi in pounds 𝑠 𝑏 𝑔 𝑛 𝑘 10 𝑠 𝑏 𝑔 𝑛 𝑏 𝑘 2 𝑔 2𝑏 𝑠 𝑏 𝑔 𝑛 𝑘 10 𝑠 𝑏 𝑔 𝑛 0 𝑏 0𝑔 0𝑛 𝑘 2 0𝑠 𝑏 2𝑔 0𝑛 0𝑘 0 From equation 4 and we can substitute this into the first three equations to reduce the 𝑔 2𝑏 system 𝑠 𝑏 2𝑏 𝑛 𝑘 10 𝑠 3𝑏 𝑛 𝑘 10 𝑠 𝑏 2𝑏 𝑛 0 𝑠 𝑏 𝑛 0 Thus we have the reduced system 𝑠 3𝑏 𝑛 𝑘 10 𝑠 𝑏 𝑛 0 𝑏 𝑘 2 Now we can solve this system step by step starting with the second equation 𝑠 𝑏 𝑛 Substituting into the first equation 𝑏 𝑛 3𝑏 𝑛 𝑘 104𝑏 2𝑛 𝑘 10 From the third equation 𝑘 2 𝑏 substituting into 4𝑏 2𝑛 2 𝑏 10 4𝑏 2𝑛 2 𝑏 103𝑏 2𝑛 8 We now have the system 3𝑏 2𝑛 8 𝑠 𝑏 𝑛 𝑘 2 𝑏 Solving this system would give the caterer the exact weights of each type of fruit she needs to buy ensuring that the salad meets the total weight and other requirements given by the recipe This would be very useful for ensuring the correct proportions of ingredients To find the 30th term you would repeatedly apply the matrix A starting with the initial vector This involves calculating powers of the matrix A and multiplying by the initial vector Computing it the number is 8869381 The inverse matrix would allow you to go backward in the sequence If you know you 𝐴 1 𝑤𝑛 can multiply by to find This is helpful if you need to find previous terms in the 𝐴 1 𝑤𝑛1 sequence from a known term Yes you could find the 5th term by using the inverse matrix and applying it multiple times 𝐴 1 starting from known terms in the sequence The 5th term in the sequence is 23 x time spent watching drama in hours y time spent watching competition shows in hours z time spent watching comedies in hours w time spent watching movies in hours He spends half as much time watching competition shows as he does watching dramas 𝑦 1 2 𝑥 His time watching sports is double his time watching dramas 𝑤 2𝑥 He spends equal amounts of time watching comedies and movies 𝑧 𝑤 The total amount of time he spends watching comedies and movies is the same as the total amount of time he spends watching dramas and competition shows 𝑧 𝑤 𝑥 𝑦 The total time he spends watching TV is 30 hours 𝑥 𝑦 𝑧 𝑤 30 𝑧 𝑤 𝑥 𝑦2𝑥 2𝑥 𝑥 1 2 𝑥 4𝑥 3 2 𝑥 𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 100 𝑥4 3𝑥2 𝑥6 1 2 𝑥2 𝑥2 3𝑥5 𝑥1 15𝑥4 𝑥2 𝑥1 9𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥1 4 5𝑥2 𝑥3 𝑥5 100 𝑥1 4 53𝑥5 𝑥3 𝑥5 100 𝑥1 13 5𝑥5 𝑥3 𝑥5 100𝑥1 14 5𝑥5 𝑥3 100 𝑥1 15𝑥4 𝑥2 153𝑥2 𝑥2 154𝑥2 60𝑥5 𝑥1 9𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 60𝑥5 93𝑥5 𝑥3 9𝑥5 𝑥 2 3 𝑥5 60𝑥5 915 5𝑥5 𝑥3 60𝑥5 139 5𝑥5 9𝑥3 9𝑥3 60𝑥5 139 5𝑥5 79 5𝑥5 Copper 𝑥1 60 Zinc 𝑥2 10 Tin 𝑥3 5 Aluminum 𝑥4 15 Nickel 𝑥5 6 Silicon 𝑥6 4 So If you double the number of bus routes each element of matrix B will be multiplied by 2 So the new matrix would be 2B The matrix 2B represents a new system where the number of bus routes has been multiplied by 2 In matrix operations multiplying by 2 means that each element in the matrix is scaled by 2 So 2B is the matrix after doubling the values in matrix B 10B means that the number of bus routes between each city is multiplied by 10 So each element in the matrix B representing the bus routes would be scaled by 10 It means there would be 10 times more bus routes between all the cities compared to the original network Its appropriate to call the matrix BS because it combines the bus routes represented by B and the subway routes represented by S This matrix shows the total number of transportation routes bus subway between the cities In this situation B represents the bus routes between cities and S represents the subway routes 4B This means that the number of bus routes between each city is multiplied by 4 2S This means that the number of subway routes between each city is multiplied by 2 Thus 4B2S represents the total number of routes between each city with bus routes quadrupled and subway routes doubled 7 Complete this graphic organizer Matrix Operations Graphic Organizer Operation Symbols Describe How to Calculate Example Using 3 3 Matrices Scalar Multiplication kA Multiply each element of matrix A by scalar k 1 2 3 4 5 6 7 8 9 2 2 4 6 8 10 12 14 16 18 The Sum of Two Matrices A B Add the corresponding elements of matrices A and B 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 10 10 10 10 10 10 10 10 10 The Difference of Two Matrices A B A 1 B Subtract the corresponding elements of matrix B from matrix A 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 8 6 4 2 0 2 4 6 8 a A B b 2A B c A C This operation is not possible because matrix A is 22 and matrix C is 33 They do not have the same dimensions so addition cannot be performed d 2C e 4D 2C f 3B 3B Since both terms are the same the result is a zero matrix 3𝐵 3𝐵 0 g 5B C This operation is not possible because matrix B is 22 and matrix C is 33 They do not have the same dimensions so subtraction cannot be performed h B 3A This operation is not possible because matrix A is 22 and matrix C is 33 The dimensions do not match so subtraction cannot be performed i C 10D j 1 C D 2 k 1 B 4 l 3D 4A This operation is not possible because matrix A is 22 and matrix D is 33 The dimensions do not match so subtraction cannot be performed m B A 1 3 2 3 a A 2B This operation is not possible because A is a 23 matrix and B is a 32 matrix The dimensions are incompatible for addition b 2A C c A C d 2C e 4D 2C This operation is not possible because D is a 32 matrix and C is a 23 matrix The dimensions are incompatible for subtraction f 3D 3D Since both terms are the same the result is a zero matrix 3𝐵 3𝐵 0 g 5B D h C 3A This operation is not possible because matrix C is 23 and matrix A is also 23 but the third dimension does not allow for this type of multiplication i B 10D j 1 C A 2 k 1 B 4 l 3A 3B This operation is not possible because A is a 23 matrix and B is a 32 matrix The dimensions are incompatible for addition m B D 1 3 2 3 The matrices C and D differ only in the 12 entry where C has 13 and D has 11 This shows that although they are not identical they are quite similar The difference likely arises from the noncommutative nature of scalar multiplication when applied before or after the addition of the fractions involved in the matrices The entry represents the total number of bus routes from City 1 to City 3 considering 𝑐13 5 both companies Since addition is commutative we know that BAAB Thus DC The entry same as in part a represents the total number of bus routes from City 1 to 𝑑13 5 City 3 when considering both companies Since CAB and DBA we observe that CD This confirms the commutative property of matrix addition The matrix A will represent Aprils Pet Supply where the element will be the number of crates 𝑎𝑖𝑗 of size i available in city j Based on the given table Aprils Pet Supply inventory This matrix represents the inventory of dog crates at Bens Pet Mart in cities 1 and 2 The third column consists of all zeroes because Bens Pet Mart does not have a store in city 3 For example represents 2 crates of Size 1 in City 1 𝑏11 2 𝑏32 1 represents 1 crate of Size 3 in City 2 So the zeros in the third column indicate that there are no crates of any size in City 3 for Bens Pet Mart To find the combined inventory when April and Ben merge their inventories we will add matrix A and matrix B Since B has only two columns for City 1 and City 2 we will assume that Aprils matrix A governs City 3 and we will add corresponding elements for City 1 and City 2 7 Jackie has two businesses she is considering buying and a business plan that could work for both Consider the tables below and answer the questions following Horuss OneStop Warehouse Supply If Business Stays the Same If Business Improves as Projected Expand to Multiple States 75 000 000 45 000 000 Invest in Drone Delivery 33 000 000 30 000 000 Close and Sell Out 20 000 000 20 000 000 Res 24Hour Distributions If Business Stays the Same If Business Improves as Projected Expand to Multiple States 99 000 000 62 500 000 Invest in Drone Delivery 49 000 000 29 000 000 Close and Sell Out 35 000 000 35 000 000 a Create matrices H and R representing the values in the tables above such that the rows represent the different options and the columns represent the different outcomes of each option H 75 000 000 45 000 000 33 000 000 30 000 000 20 000 000 20 000 000 R 99 000 000 62 500 000 49 000 000 29 000 000 35 000 000 35 000 000 b Calculate R H What does R H represent R H 99 000 000 75 000 000 62 500 000 45 000 000 49 000 000 33 000 000 29 000 000 30 000 000 35 000 000 20 000 000 35 000 000 20 000 000 R H 24 000 000 17 500 000 16 000 000 1 000 000 15 000 000 15 000 000 The matrix RH shows the difference in performance between the two businesses for each option A negative number means Horuss OneStop performs better while a positive number means Res 24Hour Distributions performs better The matrix HR shows the combined results of both businesses High positive numbers suggest favorable outcomes for both companies in those scenarios R worst 095 R 095 99 000 000 62 500 000 49 000 000 29 000 000 35 000 000 35 000 000 R worst 94 050 000 59 375 000 46 550 000 27 550 000 33 250 000 33 250 000 H best 105 H 105 75 000 000 45 000 000 33 000 000 30 000 000 20 000 000 20 000 000 H best 78 750 000 47 250 000 34 650 000 31 500 000 21 000 000 21 000 000 R best 105 R 105 99 000 000 62 500 000 49 000 000 29 000 000 35 000 000 35 000 000 R best 103 950 000 65 625 000 51 450 000 30 450 000 36 750 000 36 750 000 Graphical Solution The first line is To sketch this rewrite it as This is a line with slope 𝑥 𝑦 6 𝑦 𝑥 6 1 and yintercept at 06 The second line is Rewriting this gives This is a line with slope 3𝑥 𝑦 2 𝑦 3𝑥 2 3 and yintercept at 02 Algebraic Solution Solve the system algebraically Equation 1 𝑥 𝑦 6 Equation 2 3𝑥 𝑦 2 Subtract Equation 2 from Equation 1 𝑥 𝑦 3𝑥 𝑦 6 2 𝑥 𝑦 3𝑥 𝑦 4 4𝑥 4 𝑥 1 Substitute into Equation 1 𝑥 1 1 𝑦 6 𝑦 5 So the solution is meaning the lines intersect at the point 15 𝑥 1 𝑦 5 Graphical Solution The first line is still which is 𝑥 𝑦 6 𝑦 𝑥 6 The second line is now which is 𝑥 𝑦 2 𝑦 𝑥 2 Since they have the same slope 1 they are parallel and will not intersect Algebraic Solution Solve the system algebraically Equation 1 𝑥 𝑦 6 Equation 2 𝑥 𝑦 2 Subtract Equation 2 from Equation 1 𝑥 𝑦 𝑥 𝑦 6 2 0 4 This is a contradiction meaning the lines are parallel and do not intersect There is no solution Graphical Solution The first line is still which is 𝑥 𝑦 6 𝑦 𝑥 6 The second equation is or which is the same as 2𝑥 12 2𝑦 𝑥 6 𝑦 𝑦 𝑥 6 These are the same line so they overlap completely Algebraic Solution Rewrite the second equation as This is the same as the first equation 2𝑥 12 2𝑦 𝑥 𝑦 6 so the two lines are identical Therefore the system has infinitely many solutions No there are no other possibilities Two lines in a plane can only intersect at one point if they are not parallel and not the same line at zero points if they are parallel or at infinitely many points if they are the same line A line and a circle can Not intersect at all if the line is completely outside the circle Touch at exactly one point this is called a tangent Intersect at two distinct points if the line cuts through the circle A line can Not intersect the parabola if the line is completely above or below the curve Touch at exactly one point if the line is tangent to the parabola Intersect at two points if the line crosses the parabola Graphical Solution The circle is a circle with radius 1 centered at the origin 𝑥 2 𝑦 2 1 The line 𝑦 2𝑥 2 has slope 2 and yintercept 02 Algebraic Solution Substitute into 𝑦 2𝑥 2 𝑥 2 𝑦 2 1 𝑥 2 2𝑥 2 2 1 Simplify 𝑥 2 4𝑥 2 8𝑥 4 1 5𝑥 2 8𝑥 4 1 5𝑥 2 8𝑥 3 0 Use the quadratic formula to solve 𝑥 8 8 2453 25 8 6460 10 8 4 10 82 10 or 𝑥 6 10 0 6 𝑥 10 10 1 Find y for each x If 𝑥 0 6 𝑦 2 0 6 2 0 8 If 𝑥 1 𝑦 2 1 2 0 The line intersects the circle at the points 060806 080608 and 101 010 The equation of the circle is 𝑥 2 𝑦 2 9 The equation of the line is 𝑦 2 Substitute into the circles equation 𝑦 2 𝑥 2 2 2 9 𝑥 2 4 9 𝑥 2 5 𝑥 5 Therefore the solutions are the points 5 2 and 5 2 Substitute into the circles equation 𝑦 3 𝑥 2 3 2 9 𝑥 2 9 9 𝑥 2 0 𝑥 0 Therefore the line is tangent to the circle and intersects it at exactly one point which is 0 𝑦 3 3 Substitute y4 into the circles equation 𝑥 2 4 2 9 𝑥 2 16 9 𝑥 2 7 Since the square of a real number cannot be negative there are no real solutions for x Therefore the line y4 does not intersect the circle There are no points of intersection 𝑥 𝑦 6 𝑦 𝑥 6 Substitute this expression for y into the circle equation 𝑥 2 𝑦 2 26 𝑥 2 𝑥 6 2 26 𝑥 2 𝑥 2 12𝑥 36 26 2𝑥 2 12𝑥 36 26 2𝑥 2 12𝑥 10 0 Divide by 2 𝑥 2 6𝑥 5 0 Using the quadratic formula 𝑥 6 6 2415 21 𝑥 6 3620 2 6 16 2 𝑥 64 2 𝑥 5 or 𝑥 1 Finding y For 𝑥 5 𝑦 5 6 1 For 𝑥 1 𝑦 1 6 5 So the points where the line intersects the circle are 5 1 and 15 5𝑥 6𝑦 12 6𝑦 12 5𝑥 𝑦 125𝑥 6 Substitute this into the circle equation 𝑥 2 125𝑥 6 2 1 𝑥 2 125𝑥 2 36 1 36𝑥 2 12 5𝑥 2 36 36𝑥 2 144 120𝑥 25𝑥 2 36 61𝑥 2 120𝑥 144 36 61𝑥 2 120𝑥 108 0 Solving using the quadratic formula 𝑥 120 120 2461108 261 𝑥 120 1440026328 122 𝑥 120 11928 122 Since the discriminant is negative there are no real solutions Conclusion the line does not intersect the circle meaning there are no real solutions to the system 3𝑥 4𝑦 25 4𝑦 25 3𝑥 𝑦 253𝑥 4 Substitute this into the circle equation 𝑥 2 253𝑥 4 2 25 𝑥 2 253𝑥 2 16 25 16𝑥 2 25 3𝑥 2 400 16𝑥 2 625 150𝑥 9𝑥 2 400 25𝑥 2 150𝑥 625 400 25𝑥 2 150𝑥 225 0 𝑥 2 6𝑥 9 0 Factor so x3 𝑥 3 2 0 Finding y 𝑦 2533 4 259 4 16 4 4 Solution The line is tangent to the circle at the point 34 2𝑥 𝑦 1 𝑦 1 2𝑥 Substitute this into the circle equation 𝑥 2 1 2𝑥 2 10 𝑥 2 1 4𝑥 4𝑥 2 10 5𝑥 2 4𝑥 1 10 5𝑥 2 4𝑥 9 0 Solving the quadratic equation using the quadratic formula 𝑥 4 4 2459 25 𝑥 4 16180 10 4 196 10 𝑥 414 10 So 𝑥 1 8 or 𝑥 1 Finding the corresponding y values 𝑦 1 21 8 1 3 6 2 6 𝑦 1 2 1 1 2 3 Solution the line intersects the circle at the points 18 26 and 13 𝑥 𝑦 2 𝑦 2 𝑥 2 𝑥 𝑥 2 4 𝑥 2 𝑥 2 0 Solving the quadratic equation using the quadratic formula 𝑥 1 1 2412 21 𝑥 1 18 2 1 9 2 𝑥 13 2 So 𝑥 1 𝑜𝑟 𝑥 2 Finding the corresponding y values 𝑦 2 1 3 𝑦 2 2 0 Solution the line intersects the quadratic curve at the points 1 3 and 2 0 Since both equations represent lines set them equal to each other to find the intersection point 𝑥 𝑏 2𝑥 1 Solve for x 𝑏 2𝑥 𝑥 1 𝑏 𝑥 1 Therefore 𝑥 𝑏 1 Find the corresponding y value by substituting into either equation Using 𝑥 𝑏 1 𝑦 𝑥 𝑏 𝑦 𝑏 1 𝑏 2𝑏 1 Solution The lines intersect at the point 𝑏 1 2𝑏 1 The first equation is the equation of a circle with center and radius 2 The second equation 2 3 is a line Solve the line equation for x 𝑥 𝑦 3 Substitute this into the circle equation 𝑦 3 22 𝑦 32 4 Simplify 𝑦 12 𝑦 32 4 Expand both terms 𝑦 2 2𝑦 1 𝑦 2 6𝑦 9 4 Combine like terms 2𝑦 2 8𝑦 10 4 Simplify 2𝑦 2 8𝑦 6 0 Divide by 2 𝑦 2 4𝑦 3 0 Factor 𝑦 3𝑦 1 0 So y3 or y1 Find the corresponding x values 𝑥 𝑦 3 3 3 0 𝑥 𝑦 3 1 3 2 Solution The solutions are the points 03 and 21 Solve the first equation for x 𝑥 2𝑦 Substitute this into the second equation 2𝑦 2 2 2𝑦 𝑦 2 2𝑦 3 0 Simplify 4𝑦 2 4𝑦 𝑦 2 2𝑦 3 0 Combine like terms 5𝑦 2 2𝑦 3 0 Solve this quadratic equation using the quadratic formula 𝑦 2 2 2453 25 𝑦 2 460 10 2 64 10 𝑦 28 10 Find the corresponding x values 𝑥 20 6 1 2 𝑥 2 1 2 Solution The solutions are the points 1206 and 21 Solve the first equation for x 𝑥 4 𝑦 Substitute this into the second equation 4 𝑦 3 2 𝑦 2 2 10 Simplify 7 𝑦 2 𝑦 2 2 10 Expand both terms 49 14𝑦 𝑦 2 𝑦 2 4𝑦 4 10 Combine like terms 2𝑦 2 18𝑦 53 10 Simplify 2𝑦 2 18𝑦 43 0 Divide by 2 𝑦 2 9𝑦 21 5 0 Solve using the quadratic formula 𝑦 9 9 241215 21 𝑦 9 8186 2 9 5 2 Since the discriminant is negative there are no real solutions Conclusion There are no real solutions to this system