Topic 18: Other methods for solving systems 175 OTHER METHODS FOR SOLVING SYSTEMS Lesson 18.1 The substitution method 18.1 OPENER 1. Evaluate ab + 2c when a = 2, b = 3, and c = 5. 2. Following is a set of three shape equations. The value for each shape is constant in the three equations. Find the values for the shapes. Then explain your reasoning. + + = 15 3 = 12 = 2 a. Answer: = = = b. Here is how I figured out the answer: c. Here is how I know that my answer is correct: 18.1 CORE ACTIVITY 1. In the Opener, you figured out the values of the circle, square, and triangle in a shape equation. a. Use the same type of reasoning you used to solve the shape equation to figure out the values of x and y in the following system. y = 2x x + y = 6 Answer: x = ; y = b. Here is how I figured out the answer: 2. The swamp has a perimeter of 124 feet. The length of the swamp is 10 feet less than 5 times its width. If l = length of the swamp in feet and w = width of the swamp in feet, these equations represent the swamp situation: a. Using the answer choices provided, fill in the blanks in the following statement to describe these equations. system set four variables equations two variables Together, the two equations that model this situation form a in. of b. Just as you did in the Opener and in question 1, use substitution to solve the system representing the swamp situation.
176 Unit 6 Systems of linear equations and inequalities 3. Your class will review the solution to the Swamp Problem using the substitution method. a. Record the solution steps on the notepad image. This example will be an important reference for you as you solve linear system problems using the substitution method. b. What substitution did you make in the second equation, 2l+ 2w = 124? c. After you made this substitution into the second equation, how many variables were left in that equation? d. A common mistake when solving systems of equations is to only partially solve the system. In the example on the notepad image, you found a value for the variable w. What do you need to do now to get a full solution? e. List two ways that you can report the answer to the system of equations for the Swamp Problem. 4. In the animation you saw that you could find l by substituting the value you found for w, 12, into the equation l= 5w 10. Could you have instead substituted 12 for w in the other equation, 2l+ 2w = 124, to find l? Try it to see if you get the same result. 5. Check to make sure that the solution you found makes both equations true.
Topic 18: Other methods for solving systems 177 6. Your class will discuss the following questions. After the class reaches agreement, record the answers. a. Here is the basic idea involved with the substitution method for solving a system of equations: b. Here are the steps to take when using the substitution method for solving a system of equations: 7. Work with your partner to solve the following systems. Use the substitution method. Check your answers. y = 2x 35 6x+ 3y = 15 2x+ 3y = 10 x + 20 = 6y
178 Unit 6 Systems of linear equations and inequalities 18.1 CONSOLIDATION ACTIVITY 1. Simplify each expression by applying the Distributive Property. a. 2(3x 4) b. ( x 3) c. 5(2 7 x ) d. 1 4 (6 x 10) 2. Simplify each equation by combining like terms. a. 3x+ 2y 3x+ 6y = 24 b. 2xy + x 3y + 2xy + 3y x = 96 c. 2n 3m= 3m 2n d. 2.5a= 4.2b+ 16 4.2b+ 2.5a 3c 3. Solve each equation for the variable indicated. a. x+ y = 7 for x b. 4x+ 5y 12 = 0 for y c. 4y+ 3= 2x for x d. 3( 3x+ 2 y) = 16 + 2x for y e. PV = nrt for T
Topic 18: Other methods for solving systems 179 HOMEWORK 18.1 Notes or additional instructions based on whole-class discussion of homework assignment: 1. Why might you want to use methods other than graphing or tables to solve systems of equations? 2. Consider the following system of equations: y 2x = 4 2x+ 3y = 28 a. Solve the first equation for y. b. Now solve the system of equations using the substitution method. 3. Solve each of the following systems by substitution. a. x+ 2y = 1 b. y x = 4 x = y 5 y = 3x c. y = x y = x + 2 d. y = 5x+ 4 x + 3y = 4 e. x y = 12 x + y = 3 f. 2x+ y = 3 x + y = 5 4. Use the six-step process for solving systems of equations problems to solve the following problem. Use the substitution method. The sum of two numbers is 10 and the difference of those two numbers is 18. (Use f to represent the first number and s to represent the second number.)
180 Unit 6 Systems of linear equations and inequalities STAYING SHARP 18.1 Reviewing pre-algebra ideas 1. Lisa set a weekly goal of jogging 18 miles. On each of 3 the first 3 days, she jogged 2 miles. How many 8 more miles must she jog to reach her goal? 2. The population of Anytown is 8000. It is predicted to increase by 5% in one year. What is the predicted population after one year? Practicing algebra skills & concepts 3. Simplify each of the following expressions by applying the Distributive Property: 1( 2 x+ y ) = 1 4 c 2b = 2 3(12 5 n) = 4. Write a situation that matches the following equation. (There are many correct answers.) y = 7.50x + 2.50 Situation: Preparing for upcoming lessons 5. Matthew and Karen are shopping for vegetables to make soup. Karen buys 1 onion and 2 carrots. Her total cost (before tax) is $1.30. Matt buys 3 onions and 6 carrots. His total cost before tax is $3.90. Create two equations that you could use to find the cost of a single onion and a single carrot. Use the variable n to represent 1 onion and the variable c to represent 1 carrot. Equations: 6. For the situation in question 5, find the cost of a single carrot and the cost of a single onion.
Topic 18: Other methods for solving systems 181 Lesson 18.2 More on the substitution method 18.2 OPENER 3x+ 4y = 10 Consider the following system of linear equations: y = x 1 1. Solve the system using the substitution method. 2. Following is a graph of the two equations in the system. Explain how the graph shows the solution to the system. Note that the first equation, shown here as Y 1, has been converted to slope-intercept form. 3. Following is a table that shows the two equations in the system. Explain how the table shows the solution to the system. Note that the first equation, shown here as Y 1, has been converted to slope-intercept form. 18.2 CORE ACTIVITY 1. Solve the following systems of linear equations. Use the substitution method. Check your answer. a. y = 3x b. x = y c. x+ 2y = 60 x + y = 16 2x+ y = 5 x = y + 30
182 Unit 6 Systems of linear equations and inequalities 2. You were introduced to the following problem earlier in the unit. Set up and solve a system of linear equations to find a solution to the problem. Use the substitution method. A farmer raises chickens and cows. There are 34 animals in all. The farmer counts 110 legs on these animals. How many of each kind of animal does the farmer have? Step 1. Read the problem carefully and understand the situation. Step 2. Identify what you are looking for and assign variables. Step 3. Write equations to model the conditions in the problem. Report the two equations as a system: Step 4. Solve the system of equations. Use the substitution method. Step 5. Check the solution in both equations. Step 6. Write your answer in a sentence. 18.2 CONSOLIDATION ACTIVITY 1. Line has the equation y = 2x 4. Line m has the equation y = 5 x. a. Graph each line on the coordinate plane provided. b. Complete the following: Name a point that is on line but not on line m. Name a point that is on line m but not on line : Name a point that is on both line and line m:
Topic 18: Other methods for solving systems 183 2. Line has the equation y = 3x 4. Line m has the equation y = 3x + 2. a. Graph each line on the coordinate plane provided. b. Complete the following: Name a point that is on line but not on line m: Name a point that is on line m but not on line : Name a point that is on both line and line m: 3. Graph the following system of equations on the coordinate plane. Then state the solution to the system and check the solution. Solution: x+ 2y = 4 1 x = y+ 2 2 Check: 4. Graph the following system of equations on the coordinate plane. Then state the solution to the system and check the solution. Solution: y+ 2= 2x y 2x = 3 Check:
184 Unit 6 Systems of linear equations and inequalities 5. Line has a slope of 2 and passes through 3 the point (1,1). Line m has a slope of 2 and passes through the point (,) 03. a. Graph each line on the coordinate plane provided. b. List the intersection point of the two lines. 6. Line has a slope of 3 2. Line m has a slope of 3 2 and passes through the point( 42,) and passes through the origin. a. Graph each line on the coordinate plane provided. b. List the intersection point of the two lines. 7. Based on questions 1-6, when do two lines meet at one point? 8. Based on questions 1-6, when do two lines have no intersection point?
Topic 18: Other methods for solving systems 185 HOMEWORK 18.2 Notes or additional instructions based on whole-class discussion of homework assignment: 1. Solve the following systems of linear equations. Use the substitution method. Check your answers. Use notebook paper. a. y= x b. y= 3x+ 5 c. y= 2x+ 2 3x+ 2y= 20 y = 2x x = y + 2 d. x= 4y x + 4 = 12 e. y= x 6 2x+ y= 18 f. x+ 2y= 7 3x 2y= 3 2. Line has the equation y = 3x + 3. 1 Line m has the equation y = x 4. 2 a. Graph each line on the coordinate plane provided. b. Complete the following: Name a point that is on line but not on line m: Name a point that is on line m but not on line : Name a point that is on both line and line m: 3. You were introduced to the following problem earlier in the unit. Set up and solve a system of linear equations to find a solution to the problem. Use the substitution method. The school auditorium seats 310 people. For a particular performance, all of the seats in the auditorium are reserved. The number of seats reserved for students is 25 more than twice the amount reserved for adults (faculty, staff, and parents). How many seats are reserved for students? How many seats are reserved for adults?
186 Unit 6 Systems of linear equations and inequalities 4. Create a linear system of equations in which the two lines do not intersect. Then graph your system on the grid provided. 5. Create a linear system of equations in which the two lines intersect at one point. Graph your system of equations on the grid provided. State the intersection point and then check the solution. Solution: (, ) Check:
Topic 18: Other methods for solving systems 187 STAYING SHARP 18.2 Reviewing pre-algebra ideas 1. A rectangle in the coordinate plane has vertices at (0,0), (4,0), and (0, 2). What are the coordinates of the fourth vertex? 2. Find the sum: 3 4 + 5 9 2 2 3. What is the value of 3 x + ( 3) x when x = 1? 4. In the following set of shape equations, the value of the symbols is constant. Find the values of the shapes. Practicing algebra skills & concepts + + = 3 4 = 8 + = 5 Answer: = ; = ; = Evidence for Answer: Preparing for upcoming lessons 5. Is the following equation true? 2 2 3 3 4 4= 2 3 4 2 3 4 ( )( ) 6. Is the following equation true? (6 5)(6 5)(6 5) = ( 6 5) 3?
188 Unit 6 Systems of linear equations and inequalities
Topic 18: Other methods for solving systems 189 Lesson 18.3 Applying the substitution method 18.3 OPENER Consider each of the following situations and then state what conclusion can be drawn. 1. Fran is the same height as Jan. Fran is the same height as Nan. Conclusion: 2. The cost of tuition at East Central State College is the same as the cost of tuition at West Central State College. The cost of tuition at East Central State College is the same as the cost of tuition at North Central State College. Conclusion: 18.3 CORE ACTIVITY Here is a summary of key facts from the Hot Chocolate Problem: 15 hot chocolates (some large and some small) were ordered. A small hot chocolate costs $2. A large hot chocolate costs $3. The total amount of the order was $42. Use the six-step process and the substitution method to solve the problem. Step 2 is done for you. Step 1. Read the problem carefully and understand the situation. Step 2. Choose variables to represent the unknowns in the problem Let S = the number of small hot chocolates ordered L = the number of large hot chocolates ordered Step 3. Write two equations to model the conditions in the problem. Report the two equations as a system: Step 4. Solve the system of equations you wrote. Use the substitution method. Step 5. Check your answer. Step 6. Report your answer in the context of the problem situation. EXTENSION: Solve the problem using another system of equations solution method (tables or graphing).
190 Unit 6 Systems of linear equations and inequalities 18.3 CONSOLIDATION ACTIVITY 1. The substitution method is easier to use with some systems than others. Here are three different systems of equations. Solve each of these systems using substitution. a. 2x 5y = 18 y = 2x 10 b. x+ 5y = 4 3x+ 10y = 10 c. 2x+ 3y = 11 3x+ 2y = 4 2. Were some of them more difficult than others to solve using substitution? Why? System 2x 5y = 18 y = 2x 10 x+ 5y = 4 3x+ 10y = 10 2x+ 3y = 11 3x+ 2y = 4 Rate the difficulty level of solving using the substitution method. (Circle your choice.) Easy Medium Difficult Easy Medium Difficult Easy Medium Difficult Explanation of your rating 3. The following problem involves a system with three variables and three equations. Use the substitution method to solve this system. 3a = 12 a + b + c = 6 c = a 1 4. How can you apply reasoning similar to that which you used in the Opener to solve the following problems? Think, for example, about the Jan, Fran, and Nan height problem. (Do not solve the problems yet. Just provide an explanation to answer the question.) a. y = 2x 3 y = x + 2 b. y = 2x+ 3 y = x + 9 c. 0.5x 4 = y y = 2x 13 Explanation: 5. Apply the method you described in question 4 to solve the following problems. Show your work. (Note: We will call this method the y-equals method. You should know that this is not official mathematics terminology. Actually, the y-equals method is a special case of the substitution method. Can you see why?) a. y = 2x 3 y = x + 2 b. y = 2x+ 3 y = x + 9 c. 0.5x 4 = y y = 2x 13
Topic 18: Other methods for solving systems 191 HOMEWORK 18.3 Notes or additional instructions based on whole-class discussion of homework assignment: 1. Identify the error that was made in the following problem when applying the substitution method. Then correctly solve the problem using the substitution method. Incorrect solution: x = y+ 7 2x+ y = 8 Corrected solution: x = y+ 7 2x+ y = 8 2( y+ 7) + y = 8 2y+ 7+ y = 8 3y + 7= 8 3y = 1 1 y = 3 Description of error: x = y+ 7 1 x = + 7 3 1 22 x = 7 = 3 3 22 1 Solution:, 3 3 2. Solve the following problems using the substitution method. Use notebook paper. Here are some reminders: Consider which variable will be easier to substitute for. Consider instances where the y-equals method might be useful. a. y = 3x 8 b. x = y y = 4 x 3x+ y = 8 c. x = 4y 3x+ 2y = 28 d. 6x+ 5y = 11 y 3x = 13 e. x = y+ 8 x = y + 4 f. 2a+ b= 5 3a 2b= 4 3. Set up and solve the following problem using the substitution method. Use notebook paper. You and your friends decide to rent some studio time to make a CD. Big Notes Studio rents for $100 plus $60 per hour. Great Sounds Studio rents for $25 plus $80 per hour. Determine the number of hours for which the cost of renting the studios is the same.
192 Unit 6 Systems of linear equations and inequalities STAYING SHARP 18.3 1. In ABC, the measure of A is 108 and the measure of B is 38. What is the measure of C? 2. Find the least common multiple (LCM) of each pair of numbers: Reviewing pre-algebra ideas LCM of 3, 9: LCM of 7, 3: LCM of 6, 9: 3. What is the equation for the graph below? 4. Solve the following equation for x: Practicing algebra skills & concepts Answer: y 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 x x 2x+ 4 = 6 8 Preparing for upcoming lessons 5. A joke is being spread by a group of students. Each day, any student who has heard the joke tells it to 1 student who has not already heard it. If 2 students knew the joke at the end of the first day and 4 students knew it at the end of the second day, how many students knew the joke at the end of the third day? How many knew it at the end of the fourth day? 6. For the situation in question 5, how many students knew the joke at the end of the tenth day?
Topic 18: Other methods for solving systems 193 Lesson 18.4 The linear combination method 18.4 OPENER Use the information from Scale 1 and Scale 2 to explain why Scale 3 must balance. Scale 1 Scale 2 Scale 3 18.4 CORE ACTIVITY 1. You watched an animation that showed another method, the linear combination method, for solving linear systems problems. The system that was solved in the animation is shown here. Now solve the system yourself, using the linear combination method. x+ 2y = 12 3x 2y = 4 2. Why did the system x+ 2y = 12 3x 2y = 4 solve so easily using the linear combination method? 3. Summarize the linear combination method for solving systems of equations and then write down the steps for using this method to solve a system. a. Here is the basic idea of the linear combination method for solving a system of equations: b. Here are the steps to take when using the linear combination method for solving a system of equations: 4. Why does the linear combination method work? You can use ideas from the balance scale animation in your explanation.
194 Unit 6 Systems of linear equations and inequalities 5. Use the linear combination method to solve these systems, then answer the questions that follow. System 1 System 2 2x+ 3y = 76 2x + 5y = 20 5a 6b= 19 5a + 3b = 8 a. What makes linear combination a good method for solving these systems of equations? b. Can you think of a system for which linear combination might not be a good method? Write your system below. c. Why might it not be a good method for the system of equations that you created? 4x+ 5y = 22 6. Consider the following system of equations. 4x+ 3y = 18 a. Why doesn t adding the two equations work? b. What can you do to the equations so you can use the linear combination method to solve this system? c. Carry out your plan to find a solution for this system using the linear combination method. (Be sure to get an answer for both x and y.) 2x+ 9y = 12 7. Consider this system of equations: 4x+ 3y = 6 a. What can you do to the equations so you can use the linear combination method to solve this system? b. Carry out your plan to find a solution for this system using the linear combination method. (Be sure to get an answer for both x and y.)
Topic 18: Other methods for solving systems 195 18.4 CONSOLIDATION ACTIVITY 1. Consider this system, then answer the following questions. a. What first step could be applied to use the linear combination method to solve this system using the coefficients of the x-terms? 3x 10y = 16 18x+ 2y = 34 b. What first step could be applied to use the linear combination method to solve this system using the coefficients of the y-terms? 2. Now solve the system in question 1 using the two different methods. In the left-hand column, apply the method by manipulating the coefficients of the x-terms. In the right column, apply the method by manipulating the coefficients of the y-terms. Linear combination method solution by manipulating the coefficients of the x-terms Linear combination method solution by manipulating the coefficients of the y-terms 3. What conclusion/generalization can you make based on your work in question 2?
196 Unit 6 Systems of linear equations and inequalities 4. Think back to today s Opener. In what ways is solving the Fruit Balancing Problem similar to solving a system of linear equations problem using the linear combination method? Scale 1 Scale 2 Scale 3
Topic 18: Other methods for solving systems 197 HOMEWORK 18.4 Notes or additional instructions based on whole-class discussion of homework assignment: Use notebook paper to record your work for these problems. 1. Solve each of the following systems using the linear combination method. Check your answers. a. b. c. x+ 2y = 9 x+ y = 9 3x+ 2y = 7 x 2y = 5 x 2y = 0 x 5y = 9 d. 2x+ 7y = 5 2x+ 3y = 9 e. 8a 3b= 21 2a+ 9b= 24 f. 8x 9y = 19 4x+ y = 7 2. Identify the error that was made in the following problem when applying the linear combination method. Then correctly solve the problem using linear combination. Incorrect solution: 2x+ 3y = 24 2x y = 8 (2x+ 3 y) (2 x y) = (24 8) 2y = 16 y = 8 Description of error: Corrected solution: 2x+ 3y = 24 2x y = 8 2x + 3(8) = 24 x = 0 Solution: ( 0,8 ) 3. You were introduced to the following problem earlier in the unit. Set up and solve the problem using the linear combination method. Maggie and Mia go shopping together. At the Fashion Bee, shirts cost one price and sweaters cost one price. Maggie buys 2 shirts and 2 sweaters for $86. Mia buys 3 shirts and 1 sweater for $81. What is the cost of a shirt? What is the cost of a sweater?
198 Unit 6 Systems of linear equations and inequalities STAYING SHARP 18.4 1. How long is the rope that can be used to totally enclose the rectangular space at the beach, as shown in this diagram? 2. In this right triangle, what is the length of BC? Reviewing pre-algebra ideas Answer with supporting explanation: Practicing algebra skills & concepts 3. Complete the table, then make a graph, for the rule 1 y = x 2. 2 x Y -4-2 0 2 4 4. What is the story of the graph? Answer: Preparing for upcoming lessons 5. Jefferson was solving the system of equations shown here. He substituted y= 2x into the other equation for y, but forgot to finish solving the problem. Find the solution for Jefferson by continuing where he left off. 5x 6y= 14 and y= 2x Jefferson s work: 5x 6(2 x) = 14 6. Is the following equation true? 2 2 4 4 5 5 = (2 4 5) 3
Topic 18: Other methods for solving systems 199 Lesson 18.5 The linear combination method continued 18.5 OPENER Candi, Sandy, and Andy are working as a group to solve the linear system shown. They plan to use the linear combination method. They discuss how to do this. Which student is (or students are) correct? Explain your answer. 2a+ 3b= 18 5a 2b= 7 Candi: I don t think this system can be solved because if we want to eliminate the a terms in both equations, we can t multiply the 2 in the first equation by an integer to get a 5, which is what we would want so that when we add the equations the a terms drop out. And if we want to eliminate the b terms in both equations, we can t multiply the 2 in the second equation by an integer to get a 3, which is what we would want so that when we add the equations the b terms drop out. Sandy: But what if we multiply the first equation by 5 and the second equation by 2. Won t that cause the a terms to drop out when we add the equations? Andy: I think we can multiply the first equation by 2 and the second equation by 3. Then, when we add the equations, we ll get rid of the b terms. 18.5 CORE ACTIVITY 1. Solve the following systems of linear equations. Use the linear combination method. Be sure to check your answers. a. 4x 2y = 2 7x+ 3y = 23 b. 5x 2y = 1 4x + 9y = 14 c. 7x+ 2y = 3 5x+ 3y = 10
200 Unit 6 Systems of linear equations and inequalities 2. You were introduced to the following problem earlier in the unit. Set up and solve a system of linear equations to find a solution to the problem. Use the linear combination method. Investment A starts with $1,000 and adds $60 per week. Investment B starts with $1,500 and loses $40 per week. After how many weeks will the two investments have the same balance? Step 1. Read the problem carefully and understand the situation. Step 2. Identify what you are looking for and assign variables. Step 3. Write equations to model the conditions in the problem. Report the equations as a system: Step 4. Solve the system of equations you wrote. Use the linear combination method. Step 5. Check the solution in both equations. Step 6. Write your answer in a sentence. 18.5 CONSOLIDATION ACTIVITY 1. Ciarra and Ashley sold tickets for the high school basketball tournament. For the opening game of the tournament, the girls sold 20 student tickets and 30 non-student tickets and made $190. For the championship game, they sold 50 student tickets and 40 non-student tickets and made $300. What was the price of a student ticket? What was the price of a non-student ticket? a. Assign variables to the unknown quantities. b. Write a system of equations to model the problem.
Topic 18: Other methods for solving systems 201 c. Solve the system using the linear combination method. d. Check the answer in both equations. e. Report your answer in a complete sentence. 2. Solve the following systems using the linear combination method. Check your answers. a. b. (2,5) (21,28)
202 Unit 6 Systems of linear equations and inequalities HOMEWORK 18.5 Notes or additional instructions based on whole-class discussion of homework assignment: Use notebook paper to record your work for the problems in this homework assignment. 1. Solve each of the following systems using the linear combination method. Check your answers. a. 4x+ 2y = 12 4x + y = 18 b. x+ 3y = 17 2x+ y = 14 c. 2x+ 5y = 10 10x+ 3y = 6 d. 4x+ 2y = 14 7x 3y = 8 e. 7x+ 20y = 48 3x+ 10y = 22 f. 3x+ 5y = 11 5x+ 8y = 18 2. Earlier in the unit you solved a problem similar to the following problem. Set up and solve this problem using the linear combination method. Joseph and Patrick purchase school supplies in the school bookstore. Joseph purchases 4 notebooks and 3 pens for $11. Patrick purchases 3 notebooks and 5 pens for $11. What is the price of a notebook? What is the price of a pen?
Topic 18: Other methods for solving systems 203 STAYING SHARP 18.5 Reviewing pre-algebra ideas 1. Find the least common multiple (LCM) of each pair of numbers: LCM of 2, 8: LCM of 3, 4: LCM of 6, 8: 2. Jonathan has 7 hours to finish three jobs. If the first job 3 5 takes him 2 hours and the second job takes him 1 4 6 hours, how much time does he have to complete the third job? Practicing algebra skills & concepts 3. If you follow the Magic Number Puzzle steps below, how is the ending number related to the starting number? Directions Write down a number. Add 6. Multiply by 2. Divide by 2. Answer: How your number changes 4. Solve for y: 2(2 y+ 5) = 3y 8 Preparing for upcoming lessons 5. Use the following graph to find the coordinates of the point where the lines y = 2x+ 6 and y = 3x+ 1 intersect. y 9 8 7 6 5 4 3 2 1 x -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 6. Solve the system of equations graphed in question 5 algebraically.
204 Unit 6 Systems of linear equations and inequalities
Topic 18: Other methods for solving systems 205 Lesson 18.6 Connecting the algebra and geometry of systems of equations 18.6 OPENER Eugene and Jennifer are saving money for a summer art class. Eugene starts with $60 and adds $15 each week. Jennifer starts with $45 and adds $15 each week. 1. Write an equation for each student that represents the money saved, m, as a function of the number of weeks that have passed, w. 2. Use your graphing calculator to graph each equation. Pick an appropriate viewing window in quadrant I. Sketch your graph below and report the viewing rectangle you used. 3. Based on the equations and graphs, will Jennifer be able to catch up to Eugene in terms of money saved? If so, report when this will take place. If not, explain why. 18.6 CORE ACTIVITY 1. Solve the following system of equations using either the substitution method or the linear combination method. y = 2x+ 3 4x+ 2y = 14 Which statement (or statements) best describes what happened when you solved the system? Check all that apply. The answer includes one value for x and one value for y. Both the x and y simplified out of the equations, leaving only numbers. The numbers made a false equation, like 0 = 12. The numbers made a true equation, like 0= 0 or 18 = 18. Convert the equations in the system to slope-intercept form (write your equations in Y 1 and Y 2 ) and then graph the equations in the standard viewing rectangle. Sketch the graph in the space provided and, if it applies, state the intersection point. Intersection point: 2. Solve the following system of equations using either the substitution method or the linear combination method.
206 Unit 6 Systems of linear equations and inequalities 4x 2y = 6 2x + y = 2 Which statement (or statements) best describes what happened when you solved the system? Check all that apply. The answer included one value for x and one value for y. Both the x and y simplified out of the equations, leaving only numbers. The numbers made a false equation, like 0 = 12. The numbers made a true equation, like 0= 0 or 18 = 18. Convert the equations in the system to slope-intercept form (write your equations in Y 1 and Y 2 ) and then graph the equations in the standard viewing rectangle. Sketch the graph in the space provided and, if it applies, state the intersection point. Intersection point: 3. Solve the following system of equations using either the substitution method or the linear combination method. x+ 2y = 6 3x+ 6y = 18 Which statement (or statements) best describes what happened when you solved the system? Check all that apply. The answer included one value for x and one value for y. Both the x and y simplified out of the equations, leaving only numbers. The numbers made a false equation, like 0 = 12. The numbers made a true equation, like 0= 0 or 18 = 18. Convert the equations in the system to slope-intercept form (write your equations in Y 1 and Y 2 ) and then graph the equations in the standard viewing rectangle. Sketch the graph in the space provided and, if it applies, state the intersection point. Intersection point:
Topic 18: Other methods for solving systems 207 4. Based on these three examples, what conjectures can you make about the relationship between algebraic and geometric solutions to systems of linear equations? Complete the following table to describe your conjectures. Algebraic result Answer includes one value for x and one value for y. What does the graph look like? What might this tell you about the number of solutions for the system? Equations simplify to a false equation containing only numbers (for example, 0 = 12). Equations simplify to a true equation containing only numbers (for example, 18 = 18). 5. Create a graph of a system that is a single line. For this case, it is easier to write the system first. To see how to create such a system, look at the system in question 3. Look for a pattern in the values of the coefficients of the x-terms, the y-terms, and the constants in the two equations. Graph System of equations Algebraic solution 18.6 ONLINE ASSESSMENT Today you will take an online assessment.
208 Unit 6 Systems of linear equations and inequalities HOMEWORK 18.6 Notes or additional instructions based on whole-class discussion of homework assignment: 1. Solve each system by substitution OR linear combination. Try to make good choices about which method to use to solve each problem efficiently. Report your solution and check your answer. Write no solution or infinitely many solutions where appropriate. Use notebook paper to record your work. a. b. c. y = 5x 1 y = 15x 1 6x 3y = 6 2x y = 5 y = 3x 6 3x + y = 6 d. e. f. 4x+ 2y = 8 2x y = 2 3x+ 6y = 6 2x 3y = 4 3x+ y = 10 y = 3x + 4 2. Complete the triple-entry journal below following the directions. Directions: In the FIRST column is a list of types of solutions to system of equations problems. In the MIDDLE column, describe in your own words what happens when you solve each type of system algebraically. In the LAST column, describe in your own words what happens when you solve each type of system graphically. Type of solution to systems of equations problem a. One-solution case What happens when you solve algebraically? What happens when you solve graphically? b. No-solution case c. Infinitely many solutions case
Topic 18: Other methods for solving systems 209 3. Antonio is starring in the high school play and has lots of family members and friends who want to see him perform. He has purchased 17 tickets for the play. Some of these are adult tickets and some are child tickets. The cost of an adult ticket is $4 and the cost of a child ticket is $2. If the total cost of the 17 tickets was $60, find the number of each type of ticket purchased. a. Set up a system of equations to represent the problem. b. Solve the problem using the substitution method. c. Solve the problem using the linear combination method.
210 Unit 6 Systems of linear equations and inequalities 4. In today s lesson, you tested a conjecture about a system of equations in which the graph was a single line. Now, you will test your conjectures for the other two cases. Create a graph of a system that fits the description given. Then write equations for the system and solve the system algebraically. Do your results support your conjectures? a. Graph of the system is two lines that intersect at a single point. Graph System of equations Algebraic solution b. Graph of the system is two parallel lines. System of equations Graph Algebraic solution
Topic 18: Other methods for solving systems 211 STAYING SHARP 18.6 Reviewing pre-algebra ideas 1. Using the chart shown, what is the total distance, in miles, of biking for the five different legs of the trip? Answer with supporting work: 2. A high school play performance had two performances, one on Friday and one on Saturday. Of the tickets sold for both performances, 60% were sold for the Friday performance. If 120 tickets were sold for the Friday performance, how many were sold for the Saturday performance? Practicing algebra skills & concepts 3. What is the rule for the Input-Output table shown? Answer: Input Output -2 4-1 1 0 0 1 1 2 4 3 9 4. Change the following equation to slope-intercept form: 3x+ 2y = 5 Preparing for upcoming lessons 5. Write and solve a system of equations for the following Square Box Problem: a = b = 32 a -8 b 6. What are the coordinates of the point where the lines y = 2x+ 1 and 4x+ y = 37 intersect?
212 Unit 6 Systems of linear equations and inequalities
Topic 18: Other methods for solving systems 213 Lesson 18.7 Choosing a linear system solution method 18.7 OPENER For each of the following problems, decide what method you would use to solve the system of equations. Think about which method would be most efficient for each problem. Circle your answer choice, then write a brief explanation of why you would choose that method. You do not need to solve the system of equations. 1. y = 3x 10 y = 2x 5 Tables Graphing Substitution Linear combination Reason for choosing method: 2. 2x 3y = 18 5x+ 3y = 3 Tables Graphing Substitution Linear combination Reason for choosing method: 3. x = y 4 2x 4y = 10 Tables Graphing Substitution Linear combination Reason for choosing method: 4. 2l+ 2w = 96 l = 3w Tables Graphing Substitution Linear combination Reason for choosing method: 18.7 CORE ACTIVITY For each of the following problems, choose a method to solve the system of equations. Solve the system using that method, then explain why you chose that method. Use each method exactly once. You will need graph paper for this activity. 1. y = 2x+ 7 4x + 3y = 11 Method chosen (circle): Tables Graphing Substitution method Linear combination method State why this was a good method to use for this particular problem:
214 Unit 6 Systems of linear equations and inequalities 2. y = 3x+ 1 y = 2x + 11 Method chosen (circle): Tables Graphing Substitution method Linear combination method State why this was a good method to use for this particular problem: 3. 7x 2.5y = 1 3x+ 2.5y = 4 Method chosen (circle): Tables Graphing Substitution method Linear combination method State why this was a good method to use for this particular problem: 4. a+ b= 7 2a+ 4b= 30 Method chosen (circle): Tables Graphing Substitution method Linear combination method State why this was a good method to use for this particular problem:
Topic 18: Other methods for solving systems 215 18.7 REVIEW ONLINE ASSESSMENT You will work with your class to review the online assessment questions. Problems we did well on: Skills and/or concepts that are addressed in these problems: Problems we did not do well on: Skills and/or concepts that are addressed in these problems: Addressing areas of incomplete understanding Use this page and notebook paper to take notes and re-work particular online assessment problems that your class identifies. Problem # Work for problem: Problem # Work for problem: Problem # Work for problem:
216 Unit 6 Systems of linear equations and inequalities HOMEWORK 18.7 Notes or additional instructions based on whole-class discussion of homework assignment: Next class period, you will take the end-of-unit assessment. One good study strategy to prepare for tests is to review what you have learned. Here is a list of some of the important skills and ideas that you have worked on in this unit. Use this list to help you review these skills and concepts, especially by looking at your course materials. Another good study strategy to prepare for tests is to re-work problems that you did in class. Some specific activities to study/re-work are listed after each concept or skill. Important skills and concepts from the unit: Identify the variables and conditions in a situation and write a system of equations; Understand the meaning of a solution of a system of equations and verify a solution; Solve a system of equations using tables; Solve a system of equations using graphs by hand; Understand the importance of your mindset as a factor that can impact your motivation and learning; Solve systems of linear equations using the substitution method; Solve systems of linear equations using the linear combination method; Recognize and write the solution set for special cases of linear systems (systems in which the two lines are parallel and systems in which the two lines are collinear); connect the algebraic solution of a system of linear equations to the geometry of the case (one solution, no solution, infinite solution cases). Homework Assignment Part I: Study for the end-of-unit assessment by reviewing the key ideas listed above. Part II: Complete the online More practice in the topic Other methods for solving systems. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study. Part III: Complete Staying Sharp 18.7 As you complete the More practice, record below any questions you may have or challenges you encountered with the items.
Topic 18: Other methods for solving systems 217 STAYING SHARP 18.7 1. A stack of 500 sheets of paper is 3 inches in height. What is the thickness of one sheet of paper, in inches? Reviewing pre-algebra ideas 2. Convert 40 feet per minute to inches per second. Practicing algebra skills & concepts 2 3. Use the rule y = 3x + 2to complete the missing entries in the following table. x y -10-5 -1 0 1 5 10 4. In the following sequence, 2 is the first term, 4 is the second term, 14 is the third term, and so on. 2, -4, -14, -28, -46, -68, If n represents the term number, show that the following rule will generate the sequence: 2 2n + 4. Show: Preparing for upcoming lessons 5. Dhara graphs the functions 10x 7y = 18 and 5x 3y = 2to see where they intersect. She finds that y = 22. What value does she find for x? x y -14-13 -12-11 -10-9 -8-7 -6-5 -4-3 -2-1 -2-4 -6-8 -10-12 -14-16 -18-20 -22-24 6. After Dhara graphs the functions in question 5, she decides to use the linear combination method to verify her answer. Show how she could solve the system.
218 Unit 6 Systems of linear equations and inequalities
Topic 18: Other methods for solving systems 219 Lesson 18.8 Assessing understanding 18.8 OPENER A Fun Park refreshment stand sells only hotdogs and nachos. Hotdogs cost $2 and nachos cost $3. Marcus and his friends spent $34 at the refreshment stand. All together, they ordered a total of 14 items. How many hotdogs and how many nachos did they buy? Write a system of two linear equations in two variables that can be used to model the situation. Then choose a method to solve the system. Show your work and check your answer. 18.8 END-OF-UNIT ASSESSMENT Today you will take the end-of-unit assessment. 18.8 CONSOLIDATION ACTIVITY 1. What is the solution to the system of linear equations 2x+ 5y = 10 and 2x+ 3y = 2? 2. In the coordinate plane, what are the coordinates of the point where the lines x+ y = 1 and 2x+ y = 1 intersect? 3. If 2x y = 6 and x+ 4y = 12, what is the value of y? 4. Given 3x+ 2y = 10, what does 12x+ 8y equal? 5. Christine went to a sale at a media store. She bought 8 videos and CDs for $92. If each video cost $16 and each CD cost $10, how many of each did she buy? a. Using v to represent the number of videos that Christine bought and c to represent the number of CDs that she bought, set up a system of equations that models the information in the problem. b. Solve the system of equations and report how many of each product Christine bought. 6. For what value of a would the following system of equations have an infinite number of solutions? 2x y = 6 8x 4y = 3a
220 Unit 6 Systems of linear equations and inequalities 7. Answer the following questions to reflect on your performance and effort this unit. a. Summarize your thoughts on your performance and effort in math class over the course of this unit of study. Which areas were strong? Which areas need improvement? What are the reasons that you did well or did not do as well as you would have liked? b. Set a new goal for the next unit of instruction. Make your goal SMART. Description of goal: Description of enabling goals that will help you achieve your goal:
Topic 18: Other methods for solving systems 221 HOMEWORK 18.8 Notes or additional instructions based on whole-class discussion of homework assignment: In questions 1-3, you will apply your knowledge about systems of equations to a setting involving an amusement park. Use notebook paper to record your work for questions 1-3. 1. The admission fee at Fun Park is $13 for adults and $9 for children. On a certain day, 940 people entered the park and $10,148 was collected. How many children attended Fun Park on that day? Use a system of equations approach to find an answer. Show all work. 2. The popcorn stand sells only soft drinks and popcorn, and only one size of each. In fact, the same cup is used for both products this is part of the stand s sales pitch. A soft drink costs $2.00 and a popcorn costs $4.00. On a certain day, 120 cups were used and $330.00 was collected. How many soft drinks and how many popcorns were sold on that day? Use a system of equations approach to find an answer. Show all work. 3. Write your own Fun Park Problem. The problem should involve a system of linear equations. Then, find an answer to the problem, showing all of your work. For your Fun Park Problem, you can use the information about admissions fees and the popcorn stand given in problems 1 and 2, you can use information about Snack Shack prices given here, or you can make up your own information.
222 Unit 6 Systems of linear equations and inequalities STAYING SHARP 18.8 Reviewing pre-algebra ideas 1. Express the rate 50 yards per minute in feet per second. 2. Henry and Adam each mow lawns during the 1 summer months. Henry takes 2 hours to mow a 2 particular yard. Adam can mow the same yard in 2 1 3 hours. How much faster can Adam mow the lawn than Henry? Express your answer in minutes. Practicing algebra skills & concepts 3. Use the rule y = 3 x to complete the missing entries in the table. x y 0 1 1 3 2 9 3 4 5 6 2 4. Graph y = x 20 y 18 16 14 12 10 8 6 4 2-5 -4-3 -2-1 1 2 3 4 5-2 x Preparing for upcoming lessons 5. What are the coordinates of the point where the lines 2x+ y = 4 and 6x+ y = 10 intersect? 6. Explain how you can tell that the lines 4x 10y = 3 and 2x 5y = 1.5 are collinear by looking at the equations. Answer with reasoning: