Common Core Standard: 8.EE.8b, 8.EE.8c How can you find a rule? How can you compare two rules? How can you use what you know about solving? CPM Materials modified by Mr. Deyo
Title: IM8 Ch. 5.2.4 What if systems are not in y=mx+b form? Date: Learning Target By the end of the period, I will solve systems of equations using the Equal Values Method when equations are not in y form and learn to identify systems that represent the same line or parallel lines (that is, systems that have infinitely many solutions or no solution). I will demonstrate this by completing Four Square notes and by solving problems in a pair/group activity.
Home Work: Sec. 5.2.4 Desc. Date Due Review & Preview 3 Problems: 5 57, 5 58, 5 61
Vocabulary 1) System of Equations 2) Point of Intersection 3) Equal Values Method 4) Slope Intercept Form
5.2.4 What if systems are not in y=mx+b form? You have been introduced to systems of linear equations that are used to represent various situations. You have used the Equal Values Method to solve systems algebraically. Just as in linear equations you found that sometimes there were no solutions or an infinite number of solutions, today you will discover how this same situation occurs for systems of equations. Today you will explore a new way to approach solving a system of equations. Questions to ask your teammates today include: How can you find a rule? How can you compare two rules? How can you use what you know about solving?
5 52. Sara has agreed to help with her younger sister s science fair experiment. Her sister planted string beans in two pots. She is using a different fertilizer in each pot to see which one will grow the tallest plant. Currently, plant A is 4 inches tall and grows 2/3 inch per day, while plant B is 9 inches tall and grows 1/2 inch per day. If the plants continue growing at these rates, in how many days will the two plants be the same height? Which plant will be taller in six weeks? Write a system of equations and solve. Plant A What is the rule? y = ( )x + ( ) ( )x + ( ) = ( )x + ( ) Plant B What is the rule? y = ( )x + ( ) The two plants will be the same height after days. Plant will be the taller plant after 6 weeks (42 days).
5 53. Felipe applied for a job. The application process required him to take a test of his math skills. One problem on the test was a system of equations, but one of the equations not in y = mx + b form. The two equations are shown below. Work with your team to find a way to solve the equations using the Equal Values Method. y = 2 x 5 5 3x + 2y = 9
5 54a,b. Using the Equal Values Method can lead to messy fractions. Sometimes this cannot be avoided. But some systems of equations can be solved by simply examining them. This approach is called solving by inspection. Case I: 3x + 2y = 2 3x + 2y = 8 a) Compare the left sides of the two equations in Case I. How are they related? b) Use the Equal Values Method for solving a system of equations, write a relationship for the two right sides of the equations in Case I, and explain your result.
5 54c. Using the Equal Values Method can lead to messy fractions. Sometimes this cannot be avoided. But some systems of equations can be solved by simply examining them. This approach is called solving by inspection. Case I: 3x + 2y = 2 3x + 2y = 8 c) Graph the two equations in Case I to confirm your result for part (b) and to see how the graphs of the two equations are related.
5 54d,e. Using the Equal Values Method can lead to messy fractions. Sometimes this cannot be avoided. But some systems of equations can be solved by simply examining them. This approach is called solving by inspection. Consider the two cases below. Case II: 2x 5y = 3 4x 10y = 6 d) Recall that a coefficient is a number multiplied by a variable and that a constant term is a number alone. Compare the coefficients of x, the coefficients of y, and the two constant terms in the equations in Case II. How is each pair of integers related? e) Half of your team should multiply the coefficients and constant term in the first equation of Case II by 2 and then solve the system using the Equal Values Method. The other half of your team should divide all three values in the second equation of Case II by 2 and then solve using the Equal Values Method. Compare the results from each method. Each pair of integers related by What does your result mean?
5 54f. Using the Equal Values Method can lead to messy fractions. Sometimes this cannot be avoided. But some systems of equations can be solved by simply examining them. This approach is called solving by inspection. Case II: 2x 5y = 3 4x 10y = 6 f) Graph the two equations in Case II to confirm your result in part (e).
5 55 Additional Challenge: At the beginning of 1990, oil prices were $20 a barrel. Some oil investors predicted that the price of oil would increase by $2.25 a barrel per year. In the beginning of 2005, the price of oil was $30 a barrel. With increasing demand for oil around the world, oil investors in 2005 predicted that the price of oil would increase by $5.00 a barrel each year. a) Let x represent the number of years since 2005. Write an equation that predicts the price of oil, y, using the information available in 2005. c) Use the equations you wrote in parts (a) and (b) to determine when the cost of a barrel of oil would be the same for both price predictions. b) Investors in 1990 did not have the benefit of the 2005 information. Write an equation that represents the prediction made in 1990, using the same variables as in part (a). Remember that x represents the number of years since 2005. d) In the spring of 2011, a barrel of oil was selling for about $112. Which prediction was closer? 1990 or 2005 Was it a pretty good prediction?
5 55 Additional Challenge (WORKSPACE): At the beginning of 1990, oil prices were $20 a barrel. Some oil investors predicted that the price of oil would increase by $2.25 a barrel per year. In the beginning of 2005, the price of oil was $30 a barrel. With increasing demand for oil around the world, oil investors in 2005 predicted that the price of oil would increase by $5.00 a barrel each year.
5 56. Felipe s sister thought that he should try some more complicated systems of equations. Use what you learned in problem 5 53 to solve these two systems of equations a) x = 3 + 3y 2x + 9y = 11 b) x y = 1 x + y = 7 2
5 57a Determine the coordinates of each point of intersection without graphing. http://homework.cpm chapter/ch5/lesson/5 y = x + 8 y = x 2 The coordinates of the point of intersection are: (, )
5 57b Determine the coordinates of each point of intersection without graphing. http://homework.cpm chapter/ch5/lesson/5 y = 3x y = 4x + 2 The coordinates of the point of intersection are: (, )
5 58a,b. Change each equation below into y = mx + b form. http://hom chapter/c a) y 4x = 3 b) 3y 3x = 9
5 58c,d. Change each equation below into y = mx + b form. http://hom chapter/c c) 3x + 2y = 12 d) 2(x 3) + 3y = 0
5 59 Mailboxes Plus sends packages overnight for $5 plus $0.25 per ounce. United Packages charges $2 plus $0.35 per ounce. Mr. Molinari noticed that his package would cost the same to mail using either service. How much does his package weigh? http://hom chapter/c Mailboxes Plus United Packages The package weighs ounces.
5 60. Solve for x. http://homewo chapter/ch5/le a) 2 3 = x 4 b) 2 3 = x 4 + x 3 How are these problems similar and how are they different? These problems are similar because These problems are different because
5 61a,b Solve each equation. This problem is a checkpoint for solving equations. It will be referred to as Checkpoint 5. http://homewo chapter/ch5/l a) 3x + 7 = x 1 b) 1 2x + 5 = 4x 3
5 61c,d Solve each equation. This problem is a checkpoint for solving equations. It will be referred to as Checkpoint 5. http://homewo chapter/ch5/l c) d) 2x 6 = 2 4x (x 1) 3x 4 + 1 = 2x 5 + 5x