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Springboard 1 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Answers Teacher Copy Lesson 3-2 Solving Systems of Three Equations in Three Variables Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B p. 36 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: 1. |6| [6] 2. |–6| [6] Then have students solve the following equation. 3. |x|= 6 [x = 6 or x = –6] Example A Marking the Text, Interactive Word Wall © 2014 College Board. All rights reserved. Point out the Math Tip to reinforce why two solutions exist. Work through the solutions to the equation algebraically. Remind students that 8/25/2014 7:39 AM Springboard 2 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... 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). 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: <, 8/25/2014 7:39 AM Springboard 3 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... >, ≤, ≥, 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. 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 8/25/2014 7:39 AM Springboard 4 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... 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. Plan Pacing: 1 class period Chunking the Lesson Example A Example B #1 #2–5 #6–7 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Write the following on the board: 8/25/2014 7:39 AM Springboard 5 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... If z = 15, find the ordered pair (x, y) that satisfies the system of equations: x + y + z = −1 2x − 2y + 3z = 8 Discuss students’ answers and have them share how they solved the problem. Introduction Close Reading, Vocabulary Organizer Solving this contextual problem is not part of this activity. The problem is used simply to illustrate one type of situation that can be represented by three variables in a system of linear equations. Developing Math Language Many students have trouble visualizing a three-dimensional coordinate system when it is represented with a two-dimensional drawing. Help students understand the three-dimensional coordinate system by using the physical classroom as a model. The floor of the classroom can be thought of as Quadrant I of the xy-plane and one corner of the room can be thought of as the positive z-axis. Adjoining rooms on the same floor can be visualized to extend the xy-plane, and rooms below your classroom can be visualized to represent the negative z-axis. Example A Note Taking Point out that solving systems in three variables is similar to solving systems in two variables once a variable term has been eliminated. In this case, after the x-terms are eliminated from the three original equations, students will be solving a system in two variables—y and z. Differentiating Instruction For students who need a challenge beyond Example 3, assign the problem of finding the monetary value of the souvenir nuggets on the previous page. The correct solution is g = $0.50, s = $0.35, b = $0.25. Example B Work Backward Students often have trouble locating the source of the error when they do not arrive at the correct solution when solving a system of equations in three variables. Encourage them to write notes about what they are doing and label equations so that they can more easily retrace their steps when checking their work. Universal Access 8/25/2014 7:39 AM Springboard 6 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... For students who are struggling with solving systems in three variables in which all three variables are present in all three equations, consider beginning with the following two systems. 1. x + y + z = 4; y + z = 12; y = 6 [(4, 6, 6)] 2. x + y + z = 10; x − y = −1; x + y = 5 [(2, 3, 5)] Differentiating Instruction Challenge students to write a system of three equations in three variables that has the solution (3, −1, 5). Point out that there are an infinite number of systems with this solution. If necessary, provide the hint that students need to work backward. 1 Think-Pair-Share Have students trade flowcharts with a partner and then follow that flowchart to solve the system 2x − y − z = 10; 3x − 2y − 2z = 7; x − 3y − 2z = 10. Students can critique their partner’s flowcharts based on their effectiveness for solving the given system. Teacher to Teacher No two students seem to follow the exact same steps when solving a system of equations in three variables, which can make it difficult when students ask you for help. Have students verbally walk you through what they have done so far rather than beginning the solution process from scratch with them. It is important to model to students that understanding others’ mathematical thinking is a good practice. Differentiating Instruction Tell students that an equation in three variables represents a plane in three-dimensional space. Have students draw diagrams showing the possible ways to position three planes in three-dimensional space and then label the diagrams with the terms consistent, inconsistent, dependent, and independent. 2–5 Create Representations, Guess and Check, Sharing and Responding After students have written their equations, have them make guesses for the values of the three variables. Students should discuss their strategies for making educated rather than random guesses. 6–7 Critique Reasoning, Discussion Groups Ask students to consider and discuss the following questions: 1. Why are three equations necessary for a word problem that contains three variables? 8/25/2014 7:39 AM Springboard 7 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... 2. Would the solution to the problem still be (150, 50, 300) if the constraint from Item 4 were not included as part of the problem? Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to solving a system of equations with three variables. 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 systems of equations and solving systems of equations using Gaussian elimination and substitution. If students are having difficulty using Gaussian elimination, review strategies for reordering the equations and deciding which equation to use to eliminate variables from the other equations. Learning Targets p. 36 Solve systems of three linear equations in three variables using substitution and Gaussian elimination. Formulate systems of three linear equations in three variables to model a real-world situation. Close Reading (Learning Strategy) Definition 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 8/25/2014 7:39 AM Springboard 8 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Vocabulary Organizer (Learning Strategy) Definition Using a graphic organizer to keep an ongoing record of vocabulary words with definitions, pictures, notes, and connections between words Purpose Supports a systematic process of learning vocabulary 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 Summarizing (Learning Strategy) Definition Giving a brief statement of the main points in a text Purpose Assists with comprehension and provides practice with identifying and restating key information Paraphrasing (Learning Strategy) Definition Restating in your own words the essential information in a text or problem description Purpose 8/25/2014 7:39 AM Springboard 9 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Assists with comprehension, recall of information, and problem solving Graphic Organizer (Learning Strategy) Definition Arranging information into maps and charts Purpose Builds comprehension and facilitates discussion by representing information in visual form Group Presentation (Learning Strategy) Definition Presenting information as a collaborative group Purpose Allows opportunities to present collaborative solutions and to share responsibility for delivering information to an audience Think Aloud (Learning Strategy) Definition Talking through a difficult text or problem by describing what the text means Purpose Helps in comprehending the text, understanding the components of a problem, and thinking about possible paths to a solution Identify a Subtask (Learning Strategy) Definition Breaking a problem into smaller pieces whose outcomes lead to a solution 8/25/2014 7:39 AM Springboard 10 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Purpose Helps to organize the pieces of a complex problem and reach a complete solution Suggested Learning Strategies Close Reading, Vocabulary Organizer, Note Taking, Summarizing, Paraphrasing, Graphic Organizer, Group Presentation, Think Aloud, Identify a Subtask Sometimes a situation has more than two pieces of information. For these more complex problems, you may need p. 37 to solve equations that contain three variables. Read and discuss the material on this page with your group before you move on to Example A on the next page. Use your discussions to clarify the meaning of mathematical concepts and other language used to describe the information. With your group or your teacher, review background information that will be useful in applying concepts to the Example. In Bisbee, Arizona, an old mining town, you can buy souvenir nuggets of gold, silver, and bronze. For $20, you can buy any of these mixtures of nuggets: 14 gold, 20 silver, and 24 bronze; 20 gold, 15 silver, and 19 bronze; or 30 gold, 5 silver, and 13 bronze. What is the monetary value of each souvenir nugget? The problem above represents a system of linear equations in three variables. The system can be represented with these equations. Although it is possible to solve systems of equations in three variables by graphing, it can be difficult. Just as the ordered pair (x, y) is a solution of a system in two variables, the ordered triple(x, y, z) is a solution of a system in three variables. Ordered triples are graphed in three-dimensional coordinate space. The point (3, −2, 4) is graphed below. You can use the substitution method to solve systems of equations in three variables. Example A p. 38 8/25/2014 7:39 AM Springboard 11 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Solve this system using substitution. Step 1: Solve the first equation for z. Step 2: Substitute the expression for z into the second equation. Then solve for y. Step 3: Use substitution to solve the third equation for x. Step 4: Solve the last equation from Step 2 for y. 8/25/2014 7:39 AM Springboard 12 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Step 5: Solve the last equation from Step 1 for z. Math Tip As a final step, check your ordered triple solution in one of the original equations to be sure that your solution is correct. Solution: The solution of the system is (−4, −6, −3). Try These A Solve each system of equations using substitution. Show your work. a. (5, −1, 2) 8/25/2014 7:39 AM Springboard 13 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... b. (3, 0, −4) Math Terms When using Gaussian elimination to solve a system of three equations in the variables x, y, and z, you start by eliminating x from the second and third equations. Then eliminate y from the third equation. The third equation now has a single variable, z; solve the third equation for z. Then use the value of z to solve the second equation for y. Finally, use the values of y and z to solve the first equation for x. Another method of solving a system of three equations in three variables is called Gaussian elimination. This method has two main parts. The first part involves eliminating variables from the equations in the system. The second part involves solving for the variables one at a time. Example B p. 39 Solve this system using Gaussian elimination. Step 1: Use the first equation to eliminate x from the second equation. 8/25/2014 7:39 AM Springboard 14 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Step 2: Use the first equation to eliminate x from the third equation. Step 3: Use the second equation to eliminate y from the third equation. Step 4: Solve the third equation for z. Step 5: Solve the second equation for y. Step 6: Solve the first equation for x. 8/25/2014 7:39 AM Springboard 15 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Solution: The solution of the system is (1, 4, 2). Try These B a. Solve this system of equations using Gaussian elimination. Show your work. (−2, 5, 3) p. 41p. 40 Connect to Math History The method of Gaussian elimination is named for the German mathematician Carl Friedrich Gauss (1777–1855), who used a version of it in his calculations. However, the first known use of Gaussian elimination was a version used in a Chinese work called Nine Chapters of the Mathematical Art, which was written more than 2000 years ago. It shows how to solve a system of linear equations involving the volume of grain yielded from sheaves of rice. 1. Work with a partner or with your group. Make a flowchart on notebook paper that summarizes the steps for solving a system of three equations in three variables by using either substitution or Gaussian elimination. As you prepare your flowchart to present to the class, remember to use words and graphics that will help your classmates understand the steps. Also, be careful to communicate mathematical terms correctly to describe the application of mathematical concepts. 8/25/2014 7:39 AM Springboard 16 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Check students’ flowcharts. A farmer plans to grow corn, soybeans, and wheat on his farm. Let c represent the number of acres planted with corn, s represent the number of acres planted with soybeans, and w represent the number of acres planted with wheat. 2. The farmer has 500 acres to plant with corn, soybeans, and wheat. Write an equation in terms of c, s, and w that models this information. c + s + w = 500 3. Growing an acre of corn costs $390, an acre of soybeans costs $190, and an acre of wheat costs $170. The farmer has a budget of $119,000 to spend on growing the crops. Write an equation in terms of c, s, and w that models this information. 390c + 190s + 170w = 119,000 4. The farmer plans to grow twice as many acres of wheat as acres of corn. Write an equation in terms of c and w that models this information. 2c = w 5. Write your equations from Items 3–5 as a system of equations. 8/25/2014 7:39 AM Springboard 17 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Math Tip Determine the reasonableness of your solution. Does your answer make sense in the context of the problem? 6. Make sense of problems. Solve the system of equations. Write the solution as an ordered triple of the form (c, s, w). Explain the meaning of the numbers in the ordered triple. (150, 50, 300); The farmer should grow 150 acres of corn, 50 acres of soybeans, and 300 acres of wheat. This will meet his budget and the requirement to grow twice as many acres of wheat as corn. 7. Explain what the solution you found in Item 6 represents in the real-world situation. The farmer should plant 150 acres of corn, 50 acres of soybeans, and 300 acres of wheat. Check Your Understanding 8. Compare and contrast systems of two linear equations in two variables with systems of three linear equations in three variables. Sample answer: For consistent and independent systems, the solution of a system of two linear equations in two variables is an ordered pair, and the solution of a system of three linear equations in three variables is an ordered triple. Both types of systems can be solved by using substitution or a type of elimination. 9. Explain how you could use the first equation in this system to eliminate x from the second and third equations in the System: 8/25/2014 7:39 AM Springboard 18 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Sample answer: To eliminate x from the second equation, add the first and second equations in the system. To eliminate x from the third equation, start by multiplying the first equation by −2. Then add the resulting equation to the third equation. Lesson 3-2 Practice 10. Solve the system using substitution. (−3, 4, 0) 11. Solve the system using Gaussian elimination. (2, 2, 4) Use the table for Items 12–14. Frozen Yogurt Sales Time Period Small Cups Sold Medium Cups Large Cups Sold Sales ($) 8/25/2014 7:39 AM Springboard 19 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... Sold 1:00–2:00 6 10 8 97.60 2:00–3:00 9 12 5 100.80 3:00–4:00 10 12 4 99.20 12. Write a system of equations that can be used to determine s, m, and l, the cost in dollars of small, medium, and large cups of frozen yogurt. 13. Solve your equation and explain what the solution means in the context of the situation. (3.20, 4.00, 4.80); A small cup costs $3.20, a medium cup costs $4.00, and a large cup costs $4.80. 14. Use appropriate tools strategically. Which method did you use to solve the system? Explain why you used this method. Sample answer: I used Gaussian elimination because if I had used substitution instead, I would have ended up with equations that had fractional coefficients. 8/25/2014 7:39 AM Springboard 20 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... 8/25/2014 7:39 AM Springboard 21 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... 8/25/2014 7:39 AM Springboard 22 of 22 https://williamshartunionca.springboardonline.org/ebook/book/27E8F1B87A1C4555A1212B... 8/25/2014 7:39 AM