Chapter 4: Simultaneous Linear Equations (3 Weeks)

d. If p represents the number of pounds of peaches purchased and a represents the number ofpounds of apples purchased, the situation above can be modeled by the following equations: ! 202! 28Write in words what each of these equations represents in the context. ! 202 28e. Does the solution you found in part c) make both equations true?f. Graph the equations from part d) on the coordinate plane below. Label the lines according towhat they represent in the context.g. Find the point of intersection in the graph above. What do you notice?8WB4 - 10ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

h. The Bake Sale problem can be modeled and solved using a system of linear equations. Write inyour own words what a system of linear equations is.i. Explain, in your own words, what the solution to a system of linear equations is. How can youfind the solution in the different representations (table, graph, equation)?j. Josh really likes apple pie so he wants to donate enough money so that there are an equal numberof pounds of peaches and apples. How much does he need to donate?k. What if the students had to spend exactly 25? Exactly 20?How would the equations change?How would the graphs change?What would the new solutions be?l. What if the students wanted to make 20 pies and had exactly 64 to spend? Write the system ofequations that models this problem. Find a combination that works.8WB4 - 11ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

4.1b Class Activity: Who Will Win the Race1. Kevin and Nina are competing in a bike race. When Kevin is ninety miles into the race, he is in firstplace. Nina is in second place and is 15 miles behind Kevin.a. From this point, Kevin continues the race at a constant rate of 25 mph and Nina continues therace at a constant rate of 30 mph. When will Nina catch Kevin? Solve this problem using anymethod you aph:Other Methods:Write an Equation, Use Logic and Reasoning,Guess, Check, and Revise, Find a Pattern,Work Backwards.8WB4 - 12ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

b. If the race is 150 miles long, who will win? Assume Nina and Kevin bike at the speeds givenin part a).c. Now suppose the following: Ninety miles into the race, Kevin is still in first place and Nina isstill in second place, 15 miles behind Kevin. But now Kevin and Nina both finish out the raceat a constant speed of 30 mph. When will Nina catch Kevin? If the race is 150 miles long, whowill win?2. The graph below shows the amount of money Alexia and Brent have in savings.a. Write an equation to represent the amount y that each person has in savings after x weeks:Alexia:Brent:b. Tell the story of the graph. Be sure toinclude what the point of intersectionmeans in the context.BrentAlexia8WB4 - 13ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

4.1b Homework: Who Will Win the Race1. Gabriela and Camila like to race each other. Gabriela can run 10 feet/second while Camila can run 12feet/second. Being a good sport, Camila gives Gabriela a 20-foot head start.a. How long will it take Camila to catch Gabriela? (For ideas on how to solve this problem, seethe strategies used in the classwork)b. If the girls are racing to a tree that is 30 yards away, who will win the race? (Remember thereare 3 feet in 1 yard).2. Darnell and Lance are both saving money. Darnell currently has 40 and is saving 5 each week.Lance has 25 and is saving 8 each week.a. When will Darnell and Lance have the same amount of money?b. How much will each boy have when they have the same amount of money?c. If both boys continue saving at this rate, who will have 100 first?8WB4 - 14ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

3. The graph below shows the amount of money Charlie and Dom have in savings.a. Write an equation to represent the amount y that each person has in savings after x weeks:Charlie:Dom:b.Tell the story of the graph.CharlieDom4. Lakeview Middle School is having a food drive. The graph below shows the number of cans each classhas collected for the food drive with time 0 being the start of week 3 of the food drive.a. Write an equation to represent the number of cans y that each class hascollected after x days.Mrs. Lake’s Class: Mr. Luke’s Class:b.Tell the story of the graph.8WB4 - 15ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

4.1c Class Activity: Solving Simultaneous Linear Equations by GraphingOne method for solving simultaneous linear equations is graphing. In this method, both equations are graphedon the same coordinate grid, and the solution is found at the point where the two lines intersect.Consider the simultaneous linear equations shown below and answer the questions that follow:2 4 4 21. What problems might you encounter as you try to graph these two equations?2. What form of linear equations do we typically use when graphing?As we have seen, it is possible to rearrange an equation that is not in slope-intercept form using the same ruleswe used when solving equations. We can rearrange this equation to put it in slope-intercept form. Remember,slope-intercept form is the form !, so our goal here will be to isolate y on the left side of theequation, then arrange the right side so that our slope comes first, followed by the y-intercept.2 4Subtract 2x from both sides to isolate y 4 2 (Remember that 4 and 2 are not like terms and cannot be combined) 2 4Rearrange the right side so that the equation is truly in slope-intercept form3. Let’s look at an example that is a little more challenging. With your teacher’s help, write in the stepsyou complete as you go.4 8 16 8 16 4 " 2 #" # 24. Skill Review: Put the following equations into slope-intercept form.a. 5 9c. 4 16e. 2b. 4 2 12d. 4 2 24f. 2 5 38WB4 - 16ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

5. Consider the linear equations 2 4 and 4 2 from the previous page. Graph both equationson the coordinate plane below.a. Find the coordinates ( , ) of the point ofintersection.b. Verify that the point of intersection you foundsatisfies both equations.The solution(s) to a pair of simultaneous linear equations is all pairs (if any) of numbers ( , )that are solutions of both equations, that is ( , ) satisfy both equations. When solved graphically,the solution is the point or points of intersection (if there is one).6. Determine whether (3, 8) is a solution to the following system of linear equations:2 14 117. Determine whether (0, 5) is a solution to the following system of linear equations: 2 54 5 258WB4 - 17ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

8. Consider the equations 2 and 3. Make sure both equations are written in slope"intercept form, then graph both equations on the coordinate plane below and find the solution. Verifythat the solution satisfies both equations.!9. Consider the equations 2 1 and 2 4. Make sure both equations are written in slopeintercept form, then graph both equations on the coordinate plane below and solve the system of linearequations.10. Consider the equations 3 and 3 3 9. Graph both equations on the coordinate planebelow and solve the system of linear equations.8WB4 - 18ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

11. In the table below, draw an example of a graph that represents the different solving outcomes of asystem of linear equations:One SolutionNo SolutionInfinitely Many Solutions12. Without graphing, determine whether the following systems of linear equations will have one solution,no solution, or infinitely many solutions.a. 8 2 and 4 b. # 5 and # 1c. 2 8 and 2 2d. 5 and 2 2 10e. 3 2 5 and 3 2 6f. 2 5 and 4 2 10""13. One equation in a system of linear equations is 6 4 12.a. Write a second equation for the system so that the system has only one solution.b. Write a second equation for the system so that the system has no solution.c. Write a second equation for the system so that the system has infinitely many solutions.8WB4 - 19ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

4.1c Homework: Solving Simultaneous Linear Equations by Graphing1. Solve the system of linear equations graphically. If there is one solution, verify that your solutionsatisfies both equations.a. 3 1 and 5b. 5 and 2 3d. 2 and 2c. 3 4 and " 3!List 2 points that are solutions to this system.8WB4 - 20ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

f. 2 8 6 and 4 3e. 2 and 4!"!"Circle the ordered pair(s) that are solutions tothis system.(0, 0)g. 6 6 and 3 6(0, 1)(3, 0)(9, 3)h. 2 4 and 2 38WB4 - 21ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

2. Without graphing, determine whether the following systems of linear equations will have one solution,no solution, or infinitely many solutions.!%a. 5 and 6b. 3 9 15 and #c. 6 and 2 1#d. 5 and 53. How many solutions does the system of linear equations graphed below have? How do you know?4. One equation in a system of linear equations is 4.a. Write a second equation for the system so that the system has only one solution.b. Write a second equation for the system so that the system has no solution.c. Write a second equation for the system so that the system has infinitely many solutions.8WB4 - 22ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

5. The grid below shows the graph of a line and a parabola (the curved graph).a. How many solutions do you think there are to this system of equations? Explain your answer.b. Estimate the solution(s) to this system of equations.c. The following is the system of equations graphed above. 1 ( 2)" 1How can you verify whether the solution(s) you estimated in part b) are correct?d. Verify the solution(s) from part b).8WB4 - 23ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

4.1d Self-Assessment: Section 4.1Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box thatbest describes your progress in mastering each t MasterySubstantial MasteryUnderstanding Understanding3412I can identify theI know that I canI can graph to find the I can re-write equations in1. Solvesolutiontoafindasolutiontoasolution to a system of slope-intercept form, graphsimultaneoussystemoflinearsystemofequations, but I am not them to find the solution,linearequationswhenequationsbysure how to verifyand plug the solution backequations bygiven the graphsgraphing, but Iusing algebra that thein to verify my answergraphing.of both equations.Sample Problem#12. Understandwhat itmeans tosolve asystem ofequations.Sample Problem#13. Identify andprovideexamples ofsystems ofequationsthat have onesolution,infinitelymanysolutions, orno solution.SampleProblems #2, #34. Interpret thesolution to asystem in acontext.Sample Problem#4I know that whenI graph a systemof equations, theanswer is wherethe lines cross.I can look at thegraph of a systemof equations andtell if it has one,no, or infinitelymany solutions,but I sometimesget them mixedup.When I solve astory probleminvolving asystem ofequations Istruggle to explainwhat the solutionrepresents in thecontext.often mess upwith the graphingor getting theequations inslope-interceptform.I know that whenI graph a systemof equations, theanswer is thepoint where thetwo linesintersect, and iswritten as anordered pair (x, y).I can look at thegraph of a systemof equations andtell if it has one,no, or infinitelymany solutions. Ican sometimes telljust by looking atthe equations aswell.solution is correct.with very few mistakes.I know that thesolution to a system ofequations is the pointwhere the linesintersect, and that ifyou plug this pointinto the equations theyshould both be true.I understand that thesolution to a system ofequations is the point onthe coordinate plane wheretwo lines intersect andbecause of this, it is alsoan ordered pair thatsatisfies both equations atthe same time.I know how to tell howmany solutions a system ofequations has by lookingat a graph and by lookingat just the equations. Iunderstand what it is aboutthe structure of theequations that makes thegraphs look the way theydo. I can write a system ofequations that would haveone solution, no solution,or infinitely manysolutions.When I solve astory probleminvolving asystem ofequations, Iunderstand whatthe solutionmeans, and I canexplain tosomeone what myanswer meansmost of the time.When given a storyproblem involving asystem of equations, Ican write a sentenceexplaining what theanswer means in thecontext.I know how to tellhow many solutions asystem of equationshas by looking at agraph. I can also tellhow many solutions asystem of equationshas by looking at theequations. When givenan equation, I canwrite another equationthat would give thesystem of equationsone, no, or infinitelymany solutions.8WB4 - 24When given a storyproblem involving asystem of equations, I canwrite a sentence describingwhat the answer means inthe context. I can alsoanswer additionalquestions about thesituation.ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

Section 4.1 Sample Problems (For use with self-assessment)1. Graph the following systems of equations to find the solution. After you have found your solution,verify that it is correct. 3 51& 2Verify:' 2 7 3 9Verify:' 4 6 6 6Verify:2. Tell whether the system of equations has one solution, infinitely many solutions, or no solutions. 3 4'2 6 81 64*1 448WB4 - 25ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

3. One equation in a system of linear equations is 2 4.a. Write a second equation for the system so that the system has only one solution.b. Write a second equation for the system so that the system has no solution.c. Write a second equation for the system so that the system has infinitely many solutions.4. At the county fair, you and your little sister play a game called Honey Money. In this game she coversherself in honey and you dig through some sawdust to find hidden money and stick as much of it to heras you can in 30 seconds. The fair directors have hid only 1 bills and 5 bills in the sawdust. During thegame your little sister counts as you put the bills on her. She doesn’t know the difference between 1bills and 5 bills, but she knows that you put 16 bills on her total. You were busy counting up how muchmoney you were going to make, and you came up with a total of 40. After the activity you put the allthe money into a bag and your little sister takes it to show her friends and loses it. The fair directors finda bag of money, but say they can only give it to you if you can tell them how many 1 bills you had, andhow many 5 bills you had. What will you tell the fair directors so you can get your money back?a. Solve this problem using any method you wish. Show your work in the space below.b. Write your response to the fair directors in a complete sentence on the lines provided.8WB4 - 26ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

Section 4.2: Solve Simultaneous Linear Equations AlgebraicallySection Overview:In this section, students are solving simultaneous linear equations that have one, no, or infinitely many solutionsusing algebraic methods. The section utilizes concrete models and real world problems in order to help studentsgrasp the concepts of substitution and elimination. Students then solve systems of linear equations abstractly bymanipulating the equations. Students then apply the skills they have learned in order to solve real worldproblems that can be modeled and solved using simultaneous linear equations.Concepts and Skills to Master:By the end of this section, students should be able to:1. Determine which method of solving a system of linear equations may be easier depending on theproblem.2. Solve simultaneous linear equations algebraically.3. Create a system of linear equations to model a real world problem, solve the system, and interpret thesolution in the context.8WB4 - 27ã2014 University of Utah Middle School Math Project in partnership with theUtah State Office of Education. Licensed under Creative Commons, cc-by.

4.2a Class Activity: Introduction to SubstitutionIn the previous section, you learned how to solve a system

Chapter 4: Simultaneous Linear Equations (3 weeks) Utah Core Standard(s): Analyze and solve pairs of simultaneous linear equations. (8.EE.8) a) Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.