1 of 26 8/20/2014 2:00 PM Answers Teacher Copy Activity 3 Lesson 3-1 Systems of Linear Equations Monetary Systems Overload Solving Systems of Two Equations in Two Variables Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students should recall that an absolute value of a number is its distance from zero on a number line. Have students evaluate the following:  6  Then have students solve the following equation x = 6 College [x = 6 or Board. x = 6] All rights reserved.
2 of 26 8/20/2014 2:00 PM Example A Marking the Text, Interactive Word Wall Point out the Math Tip to reinforce why two solutions exist. Work through the solutions to the equation algebraically. Remind students that solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check their results. Developing Math Language An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two numbers that have a specific distance from zero on a number line. 1 Identify a Subtask, Quickwrite When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary. Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewise-defined functions used to write each equation and then discuss how the solution set is represented by the graph. Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler example such as: The average cost of a pound of coffee is $8. However, the cost sometimes varies by $1. This means that the coffee could cost as little as $7 per pound or as much as $9 per pound. Now have students work in small groups to examine and solve Example B by implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line. Have groups present their findings to the class. ELL Support For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think of it. Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount (greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value).
3 of 26 8/20/2014 2:00 PM Developing Math Language An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <, >,,, or. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related. Example C Simplify the Problem, Debriefing Before addressing Example C, discuss the following: Inequalities with A > b, where b is a positive number, are known as disjunctions and are written as A < b or A > b. For example, x > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x < 5 or x > 5. See graph A. This also holds true for A b. Inequalities with A < b, where b is a positive number, are known as conjunctions and are written as b < A < b, or as b < A and A < b. For example: x < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution is 5 < x < 5. See graph B. This also holds true for A b. Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the whole class. Teacher to Teacher Another method for solving inequalities relies on the geometric definition of absolute value x a as the distance from x to a. Here s how you can solve the inequality in the example: Thus, the solution set is all values of x whose distance from is greater than. The solution can be represented on a number line and written as x < 4 or x > 1.
4 of 26 8/20/2014 2:00 PM 2 Quickwrite, Self Revision/Peer Revision, Debriefing Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving inequalities. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of students present their solutions to Item 4. Assess Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Adapt Check students answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities. Activity Standards Focus In Activity 3, students write and graph systems of equations. They solve the systems of equations using graphing, substitution, and elimination. They also use technology and matrices to solve systems of equations. Throughout this activity, emphasize that there is more than one way to solve a system of equations and that some methods are more efficient in certain situations. Plan Pacing: 2 class periods Chunking the Lesson #1 2 #3 Check Your Understanding
5 of 26 8/20/2014 2:00 PM #7 Example A #11 Example B Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Have students list five solutions to the equation 2x + y = 14. Then pose and discuss the following questions: Will all students have the same five solutions? How many solutions exist for the equation? How can you visually show all of the existing solutions for the equation? Developing Math Language Be sure students understand that a solution to a system of equations is any ordered pair that, when substituted into each equation in the system, results in a true statement for every one of the equations in the system. If an ordered pair makes one equation true, but not all of the equations in the system, it is not a solution. 1 2 Shared Reading, Close Reading, Interactive Word Wall, Create Representations These first few items introduce solving systems of linear equations by graphing. Item 1 also demonstrates the limitations of graphing as a solution method. It asks students to approximate the solution by identifying a point of intersection that is not a lattice point in the coordinate plane. Review with students that a lattice point is a corner or intersection of two grid lines on the Cartesian plane. Common Core State Standards for Activity 3 HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3 Create Representations Remind students that to graph an equation, they should either write the equation in slope-intercept form or find the x- and
6 of 26 8/20/2014 2:00 PM y-intercepts. Technology Tip Students can use graphing calculators to graph each system and determine its solution. On TI calculators, the intersect option is found as option 5 under the [2nd][CALC] menu. On a TI-Nspire, this is done under the analyze option in the Graphs&Geometry tool. For additional technology resources, visit SpringBoard Digital. Developing Math Language Make sure that students understand that although there are four terms used when describing the solution set for a system of equations, there are only three classifications for a solution set: (1) inconsistent, (2) consistent and independent, (3) consistent and dependent. Mini-Lesson: Solving Systems Using a Graphing Calculator If students need additional help solving systems of equations using a graphing calculator, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to classifying a system of equations by the number of its solutions. To reinforce Item 5, have students make a sketch of the situation. 7 Predict and Confirm, Discussion Groups, Look for a Pattern Prior to using analytic geometry to solve this item, focus student attention on the starting amounts for both plans as well as the rate of change for both accounts. Students may note that they begin $3,600 apart and that the gap will narrow by $100 each month. Therefore, it will take 36 months for the accounts to be equal. Connect the initial amounts to the y-intercept and the rates of change to the slopes when solving using analytic geometry. Differentiating Instruction Tables of values can be used to answer Item 7. Creating and populating the tables of values often helps students who struggle with algebraic modeling to write equations correctly.
7 of 26 8/20/2014 2:00 PM Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to solving a system of equations by graphing. To reinforce Item 9, have students share their answers with a partner and discuss why they chose their answers. Example A Note Taking Walk students through the example. Some students may find it easier to work with whole numbers. Have them multiply the second equation by 100 to rewrite it as 2x + 5y = 20,500. Teacher to Teacher Students may struggle with this example because they fail to understand the problem. Have students solve the problem first through guess-and-check. This guess-and-check process will ensure that they understand the problem and will also motivate students to learn a more efficient way to find the solution. 11 Think-Pair-Share, Look for a Pattern Have volunteers share their answers to this item. Focus a discussion on why it is helpful to look for a variable with a coefficient of 1 first, and then, if there are no such variables, to look for a variable with a coefficient of 1 next. Example B Note Taking Work through the example with students. Refer to the Math Terms box for a summary of how to use the elimination method. Point out the importance of multiplying both sides of one equation by a number that will allow one variable term to be eliminated when the equations are added. Teacher to Teacher Students may question why they have to learn more than one way to solve a system of equations. Allow students to compare and contrast the methods by having them solve one or more of the following systems using each method Check Your Understanding Debrief students answers to these items to ensure that they understand concepts related to solving a system of equations by substitution and by elimination. Assess
8 of 26 8/20/2014 2:00 PM Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. Teacher to Teacher In Item 18, let one variable represent the number of engineers who will stay at level I, and let the other variable represent the number of engineers who will be promoted to level II. Adapt Check students answers to the Lesson Practice to ensure that they understand basic concepts related to writing systems of equations and solving systems of equations by graphing, substitution, and elimination. If students are having difficulty writing equations that model a situation, review the steps of identifying what you know and what you want to know, assigning variable names and writing equations based on what you know. Learning Targets p. 29 Use graphing, substitution, and elimination to solve systems of linear equations in two variables. Formulate systems of linear equations in two variables to model real-world situations. Shared Reading (Learning Strategy) Definition Reading the text aloud (usually by the teacher) as students follow along silently, or reading a text aloud by the teacher and students Purpose Helps auditory learners do decode, interpret, and analyze challenging text Close Reading (Learning Strategy) Definition
9 of 26 8/20/2014 2:00 PM Reading text word for word, sentence by sentence, and line by line to make a detailed analysis of meaning Purpose Assists in developing a comprehensive understanding of the text Create Representations (Learning Strategy) Definition Creating pictures, tables, graphs, lists, equations, models, and /or verbal expressions to interpret text or data Purpose Helps organize information using multiple ways to present data and to answer a question or show a problem solution Discussion Groups (Learning Strategy) Definition Working within groups to discuss content, to create problem solutions, and to explain and justify a solution Purpose Aids understanding through the sharing of ideas, interpretation of concepts, and analysis of problem scenarios Role Play (Learning Strategy) Definition Assuming the role of a character in a scenario Purpose Helps interpret and visualize information in a problem Think-Pair-Share (Learning Strategy) Definition
10 0 of 26 8/20/2014 2:00 PM Thinking through a problem alone, pairing with a partner to share ideas, and concluding by sharing results with the class Purpose Enables the development of initial ideas that are then tested with a partner in preparation for revising ideas and sharing them with a larger group Quickwrite (Learning Strategy) Definition Writing for a short, specific amount of time about a designated topic Purpose Helps generate ideas in a short time Note Taking (Learning Strategy) Definition Creating a record of information while reading a text or listening to a speaker Purpose Helps in organizing ideas and processing information Look for a Pattern (Learning Strategy) Definition Observing information or creating visual representations to find a trend Purpose Helps to identify patterns that may be used to make predictions Suggested Learning Strategies
11 1 of 26 8/20/2014 2:00 PM Shared Reading, Close Reading, Create Representations, Discussion Groups, Role Play, Think-Pair-Share, Quickwrite, Note Taking, Look for a Pattern Have you ever noticed that when an item is popular and many people want to buy it, the price goes up, but items that no one wants are marked down to a lower price? p. 30 Connect to Economics The role of the desire for and availability of a good in determining price was described by Muslim scholars as early as the fourteenth century. The phrase supply and demand was first used by eighteenth-century Scottish economists. The change in an item s price and the quantity available to buy are the basis of the concept of supply and demand in economics. Demand refers to the quantity that people are willing to buy at a particular price. Supply refers to the quantity that the manufacturer is willing to produce at a particular price. The final price that the customer sees is a result of both supply and demand. Suppose that during a six-month time period, the supply and demand for gasoline has been tracked and approximated by these functions, where Q represents millions of barrels of gasoline and P represents price per gallon in dollars. Demand function: P = 0.7Q Supply function: P = 1.5Q 10.4 Math Terms A point, or set of points, is a solution of a system of equations in two variables when the coordinates of the points make both equations true.
12 2 of 26 8/20/2014 2:00 PM To find the best balance between market price and quantity of gasoline supplied, find a solution of a system of two linear equations. The demand and supply functions for gasoline are graphed below. 1. Make use of structure. Find an approximation of the coordinates of the intersection of the supply and demand functions. Explain what the point represents. Sample answer: (9.15, 3.3); At a price of $3.30, people will demand 9.15 million gallons of gas, and companies will be willing to supply it. 2. What problem(s) can arise when solving a system of equations by graphing? Sample answer: Graphing is not very accurate if the intersection is not on a lattice point, or the scaling of the graph is not accurate enough. Technology Tip You can use a graphing calculator and its Calculate function to solve systems of equations in two variables. Math Terms Systems of linear equations are classified by the number of solutions. Systems with one or many solutions are consistent. Systems with no solution are inconsistent. A system with exactly one solution is independent.
13 3 of 26 8/20/2014 2:00 PM A system with infinite solutions is dependent. 3. Model with mathematics. For parts a c, graph each system. Determine the number of solutions. a. one solution b. no solutions c. infinitely many solutions d. Graphing two linear equations illustrates the relationships of the lines. Classify the systems in parts a c as consistent and independent, consistent and dependent, or inconsistent. a. b. c. consistent and independent inconsistent consistent and dependent Check Your Understanding p Describe how you can tell whether a system of two equations is independent and consistent by looking at its graph. If the system is independent and consistent, the graph will show a pair of lines that intersect at a point.
14 4 of 26 8/20/2014 2:00 PM 5. The graph of a system of two equations is a pair of parallel lines. Classify this system. Explain your reasoning. The system is inconsistent. A pair of parallel lines never intersect, which means that the graphs of the equations have no points in common and the system has no solutions. 6. Make sense of problems. A system of two linear equations is dependent and consistent. Describe the graph of the system and explain its meaning. The graph of the system is a single line; there are an infinite number of solutions. p. 32 Connect to Personal Finance A down payment is an initial payment that a customer makes when buying an expensive item, such as a house or car. The rest of the cost is usually paid in monthly installments. Discussion Group Tips As you work with your group, review the problem scenario carefully and explore together the information provided and how to use it to create a potential solution. Discuss your understanding of the problem and ask peers or your teacher to clarify any areas that are not clear. 7. Marlon is buying a used car. The dealership offers him two payment plans, as shown in the table. Payment Plans Plan Down Payment ($) Monthly Payment ($)
15 5 of 26 8/20/2014 2:00 PM Marlon wants to answer this question: How many months will it take for him to have paid the same amount using either plan? Work with your group on parts a through f and determine the answer to Marlon s question. a. Write an equation that models the amount y Marlon will pay to the dealership after x months if he chooses Plan 1. y = 300x b. Write an equation that models the amount y Marlon will pay to the dealership after x months if he chooses Plan 2. y = x c. Write the equations as a system of equations. Math Tip When graphing a system of linear equations that represents a real-world situation, it is a good practice to label each line with what it represents. In this case, you can label the lines Plan 1 and Plan 2. d. Graph the system of equations on the coordinate grid.
16 6 of 26 8/20/2014 2:00 PM e. Reason quantitatively. What is the solution of the system of equations? What does the solution represent in this situation? (36, 10,800); In 36 months, the total cost of both plans will be $10,800. f. In how many months will the total costs of the two plans be equal? 36 months Check Your Understanding 8. How could you check that you solved the system of equations in Item 7 correctly? Sample answer: Check that the ordered pair (36, 10,800) satisfies both of the equations in the system. 9. If Marlon plans to keep the used car less than 3 years, which of the payment plans should he choose? Justify your answer. Plan 1; The graph shows that when the time is less than 36 months (or 3 years), the total amount paid for Plan 1 is less than the total amount paid for Plan Construct viable arguments. Explain how to write a system of two equations that models a real-world situation. Sample answer: Identify the two quantities in the situation that can vary. Assign variables to these quantities. Write an equation in terms of the two variables that models part of the situation. Then write a second equation in terms of the two variables that models another part of the situation. Finally, write the two equations as a system.
17 7 of 26 8/20/2014 2:00 PM p. 33 Math Terms In the substitution method, you solve one equation for one variable in terms of another. Then substitute that expression into the other equation to form a new equation with only one variable. Solve that equation. Substitute the solution into one of the two original equations to find the value of the other variable. Investors try to control the level of risk in their portfolios by diversifying their investments. You can solve some investment problems by writing and solving systems of equations. One algebraic method for solving a system of linear equations is called substitution. Example A During one year, Sara invested $5000 into two separate funds, one earning 2 percent and another earning 5 percent annual interest. The interest Sara earned was $205. How much money did she invest in each fund? Step 1: Let x = money in the first fund and y = money in the second fund. Write one equation to represent the amount of money invested. Write another equation to represent the interest earned. Step 2: Use substitution to solve this system.
18 8 of 26 8/20/2014 2:00 PM Step 3: Substitute the value of x into one of the original equations to find y. Math Tip Check your answer by substituting the solution (1500, 3500) into the second original equation, 0.02x y = 205 Solution: Sara invested $1500 in the first fund and $3500 in the second fund. Try These A Write your answers on notebook paper. Show your work. Solve each system of equations, using substitution. Check students work.
19 9 of 26 8/20/2014 2:00 PM a. ( 14, 13) b. (12, 1) c. (3, 7) d. Model with mathematics. Eli invested a total of $2000 in two stocks. One stock cost $18.50 per share, and the other cost $10.40 per share. Eli bought a total of 130 shares. Write and solve a system of equations to find how many shares of each stock Eli bought. p. 34
20 0 of 26 8/20/2014 2:00 PM 11. When using substitution, how do you decide which variable to isolate and which equation to solve? Explain. Sample answer: Choose a variable that is easy to isolate by finding the equation with a variable that has a coefficient of 1 or 1. Another algebraic method for solving systems of linear equations is the elimination method. Example B p. 35 Math Terms In the elimination method, you eliminate one variable. Multiply each equation by a number so that the terms for one variable combine to 0 when the equations are added. Then use substitution with that value of the variable to find the value of the other variable. The ordered pair is the solution of the system. The elimination method is also called the addition-elimination method or the linear combination method for solving a system of linear equations. A stack of 20 coins contains only nickels and quarters and has a total value of $4. How many of each coin are in the stack? Step 1: Let n = number of nickels and q = number of quarters. Write one equation to represent the number of coins in the stack. Write another equation to represent the total value.
21 1 of 26 8/20/2014 2:00 PM Step 2: To solve this system of equations, first eliminate the n variable. Step 3: Find the value of the eliminated variable n by using the original first equation. Step 4: Check your answers by substituting into the original second equation. Solution: There are 5 nickels and 15 quarters in the stack of coins. Try These B Solve each system of equations using elimination. Show your work. Check students work. a. (5, 5)
22 2 of 26 8/20/2014 2:00 PM b. (2, 4) c. ( 3, 4) d. A karate school offers a package of 12 group lessons and 2 private lessons for $110. It also offers a package of 10 group lessons and 3 private lessons for $125. Write and solve a system of equations to find the cost of a single group lesson and a single private lesson. Check Your Understanding 12. Compare and contrast solving systems of equations by using substitution and by using elimination. Sample answer: In both methods, you start by solving for the value of one of the variables and then use that value to solve for the value of the other variable. In the substitution method, you use substitution to get rid of one of the variables. In the elimination method, you add equations to get rid of one of the variables.
23 3 of 26 8/20/2014 2:00 PM 13. Reason abstractly. Ty is solving the system using substitution. He will start by solving one of the equations for x. Which equation should he choose? Explain your reasoning. Sample answer: The first equation; to solve the first equation for x, Ty only needs to add 2y to both sides, but to solve the second equation for x, Ty would need to do two steps: first, subtract 6y from both sides, and then divide both sides by Explain how you would eliminate one of the variables in this system: Sample answer: Multiply the second equation by 2 to get. Then add the equations to eliminate the variable y and get 8x = 33. Lesson 3-1 Practice 15.Solve the system by graphing. ( 2, 5) 16.Solve the system using substitution.
24 4 of 26 8/20/2014 2:00 PM ( 5, 6) 17.Solve the system using elimination. (3, 4) 18. Make sense of problems and persevere in solving them. At one company, a level I engineer receives a salary of $56,000, and a level II engineer receives a salary of $68,000. The company has 8 level I engineers. Next year, it can afford to pay $472,000 for their salaries. Write and solve a system of equations to find how many of the engineers the company can afford to promote to level II., where x is the number of engineers who will stay at level I and y is the number of engineers who will be promoted to level II; solution: (6, 2); The company can afford to promote 2 engineers to level II. 19. Which method did you use to solve the system of equations in Item 18? Explain why you chose this method. Answers will vary.
of 19 9/2/2014 12:09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students
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