CHAPTER 1: SYSTEMS OF LINEAR EQUATIONS

Transcription

1 CHAPTER 1: SYSTEMS OF LINEAR EQUATIONS Specific Expectations Addressed in the Chapter Solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination. [1.4, 1.5, 1.6, 1.7, Chapter Task] Solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method. [1.3, 1.4, 1.6, 1.7, Chapter Task] Prerequisite Skills Needed for the Chapter Graph a linear relation using the x- and y-intercepts. Graph a linear relation using the slope and y-intercept. Identify a relation as linear or nonlinear, given an equation or a table of values. Identify a direct or partial variation, given an equation or a graph. Solve a linear equation in one variable. Solve a linear equation for one variable after substituting a known value for the other variable. Expand and simplify an algebraic expression. What big ideas should students develop in this chapter? Students who have successfully completed the work of this chapter and who understand the essential concepts and procedures will know the following: The solution to a linear system with two linear equations, where one or both have two variables, consists of the values that make both equations true. A pair of linear equations in x and y can be solved by graphing the equations and determining the point of intersection. A pair of linear equations in x and y can be solved by substitution: solving one equation for one variable and substituting this expression into the other equation. Different linear systems can be equivalent; that is, they can have the same solution. A pair of linear equations in x and y can be solved by elimination: creating an equivalent system with either the x-coefficients or the y-coefficients the same or opposite. A pair of linear equations in x and y has no solutions when the graphs of the equations are parallel and not identical, one solution when the graphs are not parallel, or infinitely many solutions when the graphs are identical. Chapter 1 Introduction 1

2 Chapter 1: Planning Chart Lesson Title Lesson Goal Pacing 13 days Materials/Masters Needed Getting Started, pp. 4 7 Use concepts and skills developed prior to this chapter. 2 days grid paper; ruler; Diagnostic Test Lesson 1.1: Representing Linear Relations, pp Use tables, graphs, and equations to represent linear relations. 1 day grid paper; ruler; graphing calculator; Lesson 1.1 Extra Practice Lesson 1.2: Solving Linear Equations, pp Connect the solution to a linear equation and the graph of the corresponding relation. 1 day grid paper; ruler; graphing calculator; Lesson 1.2 Extra Practice Lesson 1.3: Graphically Solving Linear Systems, pp Use graphs to solve a pair of linear equations simultaneously. 1 day grid paper; ruler; graphing calculator; Lesson 1.3 Extra Practice Lesson 1.4: Solving Linear Systems: Substitution, pp Solve a system of linear equations using an algebraic strategy. 1 day grid paper; ruler; graphing calculator; Lesson 1.4 Extra Practice Lesson 1.5: Equivalent Linear Systems, pp Compare solutions for equivalent systems of linear equations. 1 day grid paper; ruler; Lesson 1.5 Extra Practice Lesson 1.6: Solving Linear Systems: Elimination, pp Solve a linear system of equations using equivalent equations to remove a variable. 1 day Lesson 1.6 Extra Practice Lesson 1.7: Exploring Linear Systems, pp Mid-Chapter Review, pp Chapter Review, pp Chapter Self-Test, p. 64 Curious Math, p. 29 Chapter Task, p. 65 Connect the number of solutions to a linear system with its equations and graphs. 1 day graphing calculator, or grid paper and ruler 4 days grid paper; ruler; graphing calculator; Mid-Chapter Review Extra Practice; Chapter Review Extra Practice Chapter Test 2 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

3 CHAPTER OPENER Using the Chapter Opener Introduce the chapter by discussing the photograph on pages 2 and 3 of the Student Book, relating it to the red and blue graphs at the bottom of page 3. Guide students to talk about how the red and blue graphs are similar and how they are different. Questions you might ask include the following: Which parts of the graphs can you identify, and what do these parts tell you? (For example, the y-intercepts are the costs of the bulbs; the intersection point is where the bulbs have the same cost for the same number of hours of use.) Are incandescent bulbs really free, or are they just very cheap? (Elicit from students how enlarging the vertical scale would show prices more exactly.) What questions could you ask about the graph? The photograph shows a cityscape at night. Connect the photograph to the graphs and to the question about lighting costs on page 3 by asking students to estimate the cost of keeping every light visible in the photograph burning for 1 h. Encourage students to suggest strategies to answer the question, and work through the strategies with the class. One strategy would be to estimate the number of light bulbs, which would give the number of bulb-hours, and then use the graph on page 3 to estimate the cost of all-incandescent vs. all-fluorescent lighting. Read the goals with the class, relating them to what students learned in Grade 9 math, where possible. Chapter 1 Opener 3

4 GETTING STARTED Using the Words You Need to Know Student Book Pages 4 7 Students might read each term and place it in the given sentences, look at the sentences and search for the matching terms, eliminate choices by matching what they are sure they know, or use a combination of strategies. If students are unsure of the definition of a term, suggest that they look up the term in the Glossary and then write the term and definition in their notes. After students have completed the question, ask them to use the terms in other sentences as pairs or as a class. Using the Skills and Concepts You Need Work through each of the examples in the Student Book (or similar examples, if you would like students to see more examples), and invite students to ask questions about the solution. Discuss the different tools and strategies for the first example, having students demonstrate the use of a graphing calculator. You could make algebra tiles available for students to demonstrate the second example. Ask students to look over the Practice questions to see if there are any questions they do not know how to solve. Refer students to the Study Aid chart in the margin of the Student Book for more help. Allow students to work on the Practice questions in class, and assign any unfinished questions for homework. Preparation and Planning Pacing 5-10 min Words You Need to Know min Skills and Concepts You Need min Applying What You Know Materials grid paper ruler Nelson Website Using the Applying What You Know Have students work individually on the activity. Have them read the whole activity before beginning their work. Ask them to make predictions about the answers for parts D and E before doing parts A through C. Make sure that they choose values for the table that will help confirm their predictions. After students complete the activity, ask for the greatest and least numbers of each type of bill that Barb could receive. Discuss various ways to represent the combinations in part H: using the graph, creating the table, and using logical reasoning. Answers to Applying What You Know A. He gives Barb 12 $5 bills. B. Answers may vary, e.g., Number of $5 Bills Number of $10 Bills Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

5 C. 5x + 10y = 100 D. Use a broken line, because not all points on the line are possible values of x and y. E. The number of $10 bills decreases as the number of $5 bills increases. F. The x-intercept represents the number of $5 bills if there were no $10 bills. The y-intercept represents the number of $10 bills if there were no $5 bills. G. Any points whose coordinates are not both positive integers or zero are not possible combinations. H. The possible combinations are 0 $5 bills and 10 $10 bills, 2 $5 bills and 9 $10 bills, 4 $5 bills and 8 $10 bills, 6 $5 bills and 7 $10 bills, 8 $5 bills and 6 $10 bills, 10 $5 bills and 5 $10 bills, 12 $5 bills and 4 $10 bills, 14 $5 bills and 3 $10 bills, 16 $5 bills and 2 $10 bills, 18 $5 bills and 1 $10 bill, 20 $5 bills and 0 $10 bills. Initial Assessment When students understand What You Will See Students Doing If students misunderstand Students construct a table with correct pairs of values. Students write a correct equation by forming expressions for the value of each set of bills and by recognizing that the total value of both sets of bills is constant. Students graph the equation or coordinates in the table, labelling the axes correctly. Students determine all the possible combinations of bills correctly. Students may include incorrect pairs of values, with sums that are not $100, in their table. Some students may misunderstand the relation between the number of $5 bills and the number of $10 bills. Students are unable to write an equation, or they may write an equation in which the coefficients of x and y do not result in $100 for all possible values. They may reverse x and y so that x does represent the number of $5 bills and y does not represent the number of $10 bills, or they may not record $100 for the sum of the values. Students may not draw a straight graph with the correct intercepts or through the correct points. The graph may not be a broken line. Students may not label the axes appropriately. Students may not interpret the coordinates of the graph that are positive integers or zero as the possible combinations. They may not realize why the number of $5 bills cannot be an odd number or why the coordinates with 0 need to be included (since it is possible to have only $5 bills or only $10 bills). Chapter 1 Getting Started 5

6 1.1 REPRESENTING LINEAR RELATIONS Lesson at a Glance GOAL Use tables, graphs, and equations to represent linear relations. Prerequisite Skills/Concepts Graph a linear relation using the x- and y-intercepts. Graph a linear relation using the slope and y-intercept. Identify a relation as linear or nonlinear, given an equation or a table of values. Identify a direct or partial variation, given an equation or a graph. Mathematical Process Focus Reasoning and Proving Selecting Tools and Strategies Connecting Student Book Pages 8 14 Preparation and Planning Pacing 5 10 min Introduction min Teaching and Learning min Consolidation Materials grid paper ruler graphing calculator Recommended Practice Questions 5, 6, 7, 10, 11, 12, 14 Key Assessment Question Question 12 Extra Practice Lesson 1.1 Extra Practice Nelson Website MATH BACKGROUND LESSON OVERVIEW In this lesson, students explore representations of linear relations: equations, tables of values, and graphs. Students connect linear relations to problem situations, interpreting values in the relations as solutions. 6 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

7 1 Introducing the Lesson (5 to 10 min) Research some cell-phone plans before beginning this lesson, and present them as the basis for an introductory discussion. Alternately, ask students to do the research or to describe cell-phone plans they know about. Ask the class questions such as these: How does good value in a cell-phone plan depend on how the phone is used? When might pay-as-you-go plans be good value? What features do you consider important for a plan? Why are these features important? 2 Teaching and Learning (20 to 25 min) Learn About the Math Example 1 demonstrates how to create an algebraic model for a simple partial variation situation. The emphasis is on exploring possible combinations that satisfy the model. Guide the students through the solution, having them explain Aiko s thinking and Malcolm s thinking. Ask: How do you know that Aiko and Malcolm represented the same information? Technology-Based Alternative Lesson If TI-nspire calculators are available, students can use them for the examples: For Example 1, students can use a table to show the possible results. You might need to remind them how to enter a formula into a spreadsheet as described in Appendix B-42. When students determine an equation for a given situation such as for Example 2, have them write the equation in the form y = mx + b and enter it as a formula. Then, to show that the relationship is linear, they can determine the first differences using the List feature as described in Appendix B-43. A table for the information in Example 2 might look like the one at the right. For Example 3, students may need to compare two equations to see which one is better for a given situation. Students can enter each equation (in the form y = mx + b) in the Graphs & Geometry application. Then they can use either the Trace feature to trace along each line or the Point(s) of Intersection feature as described in Appendix B-47 to determine the point of intersection. If students use the Trace feature, they will need to move the point as close as possible to the intersection. 1.1: Representing Linear Relations 7

8 Answers to Reflecting A. The differences between rows are constant for both text messages and minutes. B. Malcolm determined the x-intercept of the graph by setting y = 0 and then solving for x. He determined the y-intercept of the graph by setting x = 0 and then solving for y. C. Answers may vary, e.g., Aiko would find the table more useful because it shows the values for combinations of messages and minutes. Aiko would find the graph more useful because she could easily see the amount of minutes she can afford for any number of messages, and vice versa. 3 Consolidation (30 to35 min) Apply the Math Using the Solved Examples Example 2 demonstrates the use of a graphing calculator to model a partial variation, which is the same type of variation shown in Example 1. Work through the example with the class. Ensure that students think about how the terms 2x and 1.50y are connected to the problem and what they need to do with the equation in its original form to enter it into their calculators. Example 3 compares two linear models, as preparation for the main theme of the chapter. You might have students work in pairs, with one partner explaining the models and the other partner entering the models into the calculator. Check that students can use the window settings and Trace feature correctly, directing them to the material in Appendix B or guiding them through it if necessary. Answer to the Key Assessment Question When discussing answers for question 12, ask students what they know about interest for savings accounts and for government bonds. Ensure that students realize that they can use different variables. Choices may be s for the amount in the savings account, and g or b for the amount in a government bond. 12. a) Let x represent the amount, in dollars, invested in the savings b) account. Let y represent the amount, in dollars, invested in the government bond. The savings account earns 3% interest, or 0.03x. The bond earns 4% interest, or 0.04y. Since the total interest earned is $150, the equation is 0.03x y = Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

9 Closing After students read question 14, ensure that they understand what is meant by Non-examples : relations that are not straight lines. Students could work with a partner or on their own. Provide an opportunity for students to share their charts, for example, by posting their charts or discussing their charts in groups. Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students form correct terms to represent the elements of a problem situation. Students use the x- and y-intercepts of a linear relation to graph the line. Students use the slope-intercept form of a linear relation to graph the line with and without technology. Students use different representations of a linear relation to reason successfully about a problem. If students misunderstand Students may be unable to identify the variables in the problem situation, or they may form a mathematical expression that does not accurately represent the situation. Students cannot calculate the x- and y-intercepts of a linear relation correctly, or they are unable to plot the intercepts correctly. They may not set the appropriate variable equal to zero to solve for the other variable. Students cannot rearrange a linear equation to the slopeintercept form, or they cannot use the slope-intercept form to graph the line correctly with or without technology. Students can use only one representation of a linear relation to reason about a problem, they reason incorrectly, or they cannot relate the representations. Key Assessment Question 12 Students write an algebraic equation for the relation to describe the amounts earned by the two investments. Students graph the relation using any appropriate strategy. Differentiating Instruction How You Can Respond EXTRA SUPPORT Students cannot express the amount for each interest rate correctly or write the correct algebraic expression for the sum of the investments. They may not be able to relate the sum to the total amount earned. Students cannot apply an appropriate strategy to graph the relation and/or graph it incorrectly. They may not be able to determine the intercepts or the slope and y-intercept. 1. When working with application problems, students may have difficulty interpreting the information given. Some students may find it easier to describe the situation in a sentence or draw a picture or graph of the information before writing the equation. 2. If students are having difficulty graphing equations in slope-intercept form, explain that they can plot the y-intercept first and 2 then use rise over run (e.g., if m =, move 2 right and 3 down) to locate other points on the line. 3 EXTRA CHALLENGE 1. Have students work in pairs to create an equation of the form Ax + By = C, without discussing its meaning. Then have them create a problem situation that could give rise to their equation. 1.1: Representing Linear Relations 9

10 1.2 SOLVING LINEAR EQUATIONS Lesson at a Glance GOAL Connect the solution to a linear equation and the graph of the corresponding relation. Prerequisite Skills/Concepts Solve a linear equation in one variable. Solve a linear equation for one variable after substituting a known value for the other variable. Create a linear model to represent a situation. Mathematical Process Focus Problem Solving Selecting Tools and Computational Strategies Representing Student Book Pages Preparation and Planning Pacing 5-10 min Introduction min Teaching and Learning 35 min Consolidation Materials grid paper ruler graphing calculator Recommended Practice Questions 4, 6, 8, 10, 11, 13 Key Assessment Question Question 10 Extra Practice Lesson 1.2 Extra Practice Nelson Website MATH BACKGROUND LESSON OVERVIEW The topic of solving linear equations is introduced with a problem that involves a single linear equation of the form y = mx + b. This type of problem may also be viewed as a linear system of two equations and two variables, where the equations are in the form y = mx + c and y = constant. (See the graphing-calculator solution in Example 2.) 10 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

11 1 Introducing the Lesson (5 to 10 min) Have students work in pairs to record up to 15 words about representations of linear relations, using point form, lists, and sentences. Examples of words are straight graph, dotted or broken line, constant first differences, intersect y-axis unless vertical, intersect x-axis unless horizontal, equal slope between points. Ask a few pairs to read their words. Then ask if anyone has words that have not yet been mentioned. Have these words read. 2 Teaching and Learning (20 to 25 min) Learn About the Math Example 1 introduces the theme of the lesson, which is constructing a linear relation in the form of an equation in one variable, substituting a target or budget value for this variable, and solving for the other variable. Read the situation for Example 1 with the class, and lead students informally through the first solution by asking questions such as these: If Joe spends $9.95 on the monthly fee, how much does he have left for downloads? If Joe downloads 50 songs, how much does he spend on downloads? Discuss the advantages of each of the three strategies with the class. For example, the first strategy may seem to be the most straightforward, but if a different monthly budget is set, one of the other strategies would probably be easier. Technology-Based Alternative Lesson If TI-nspire calculators are available, have students enter the equation that models the situation in Example 1 into the Graphs & Geometry application. Then they can trace the graph until they get as close to $40 as possible. They could also place a point on the line and then change the y-value to 40 as shown in the screen at the right. Placing a point on the line is described in Appendix B-39. The point would move to this location, and students could then read the x-value that produces $40. (If students choose this strategy, you may need to remind them that if the desired y-value is not in the current window, the point will not move to this location.) Another way to look at the table of values that corresponds to the relation is to add a Function Table to the screen as described in Appendix B-41 and B-42. If this table is too small, then add a Lists & Spreadsheet application and switch to Function Table. If desired, students can change the starting value of x and the increment shown in the table by editing the Function Table Settings. 1.2: Solving Linear Equations 11

12 Answers to Reflecting A. Similarity: Both solutions involve subtracting $9.95 from $40 and dividing the result by $0.55. Difference: Tony s solution involves writing an equation with a variable and substituting $40 for one of the variables, but William s solution presents calculations for specific values only. B. The x-coordinate of the point closest to y = 40, but less than y = 40, represented the maximum number of songs. The y-coordinate of this point represented Joe s budget. C. Answers will vary, e.g., I prefer William s strategy because it is a straightforward, common-sense solution. I prefer Tony s strategy because it uses an algebraic model, which can be solved for n as desired. As well, the algebraic model can be adapted for different values of C. I prefer Lucy s strategy because the relation between the cost and the number of downloads can be seen as a graph. Lucy s strategy can be adapted for different values of C. 3 Consolidation (35 min) Apply the Math Using the Solved Examples In Example 2, a linear equation is created to model a problem that involves a decreasing quantity. In Stefani s solution, the equation is solved algebraically after substituting for the amount of fuel remaining. In Henri s solution, the equation is graphed, along with another equation of the form y = constant. The solution is then determined using the Intersect operation. Answer to the Key Assessment Question For question 10, students can write the equation C = d, where C is the cost in dollars and d is the distance in kilometres, substitute C = 90 since the budget is $90, solve for d to obtain the result of about km, and then round to the exactness of the given information. Students could use a graphing calculator to check their answers. 10. Maria and her grandmother can travel about 208 km. Closing If students need help with question 13, refer to the examples. Looking forward to Lesson 1.3, it may be helpful for students to realize that, in addition to reading the x-coordinate from the graph for the given y-coordinate, they can view the given value of y as a second (horizontal) line and the solution as an intersection point. 12 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

13 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students correctly write linear equations to model problems. Students correctly convert a linear equation in two variables to a linear equation in one variable by substitution. Students correctly solve a linear equation in one variable by applying algebraic and graphical methods. Students demonstrate the correct use of technology, such as graphing calculators, to solve problems that involve a linear equation. If students misunderstand Students may be unable to identify the variables, or they may misinterpret the given information and write inaccurate equations to model problems. Students substitute for the wrong variable, or they substitute the wrong value. Students cannot apply inverse operations correctly, or they do not know how to interpret a graph to determine the solution. Students may be unable to enter equations correctly, determine appropriate window settings, or use the calculator features to locate points of intersection or points with a given coordinate. Key Assessment Question 10 Students write an algebraic equation to represent the relation between distance and cost. Students correctly substitute the budget amount into their equation and solve for the round-trip distance. Students may introduce a variable for the number of days, or they may try to write an equation to represent distance rather than cost. Students substitute the budget amount for the wrong variable, or substitute the wrong amount, or are unable to use inverse operations to solve for the round-trip distance. Differentiating Instruction How You Can Respond EXTRA SUPPORT 1. Students may find it helpful to record their thinking, or to explain their thinking to a partner, as they solve a problem. The student thinking boxes at the right of each solution can be used as guidance. Students might start with the problems that are most similar to the examples. 2. If students have difficulty solving equations, explain how it is necessary to perform the same operation on both sides of the equation to keep the sides equal. 3. To help students understand solutions with calculators, have them explore a problem by substituting various trial x-values and looking at the corresponding y-values. You can then point out that if they graph (or tabulate) the relation, they will be able to see all these results at once. (They may need guidance through the window settings or table settings.) EXTRA CHALLENGE 1. Have students create problems that could be solved using linear equations and then exchange problems with a partner. 2. Ask students to select an equation in the lesson and explain how the value of one variable affects the other variable. For example, when one variable increases, what happens to the other variable? Have them explain how this makes sense for the algebra, the graph, and the situation. 1.2: Solving Linear Equations 13

14 1.3 GRAPHICALLY SOLVING LINEAR SYSTEMS Lesson at a Glance GOAL Use graphs to solve a pair of linear equations simultaneously. Prerequisite Skills/Concepts Graph a linear relation using the x- and y-intercepts Graph a linear relation using the slope and y-intercept. Create a linear model to represent a situation. Interpret the solution to a linear equation in terms of the graph of the corresponding relation. Specific Expectation Solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate [algebraic or] graphical method. Mathematical Process Focus Problem Solving Selecting Tools and Computational Strategies Representing Student Book Pages Preparation and Planning Pacing 5 min Introduction min Teaching and Learning min Consolidation Materials grid paper ruler graphing calculator Recommended Practice Questions 4, 7, 10, 12, 13, 15, 16 Key Assessment Question Question 13 New Vocabulary system of linear equations solution to a system of linear equations Extra Practice Lesson 1.3 Extra Practice Nelson Website MATH BACKGROUND LESSON OVERVIEW Students develop the understanding that just as one linear equation in one variable may have a solution (a value that makes it true), two linear equations in two variables may have two solutions. They also learn that the two-equation, two-variable format is an example of a linear system. This concept is explored throughout the rest of Chapter 1. Students apply representation skills from Lesson 1.1 to the conceptually simplest strategy for solving a linear system, which is graphing. In later lessons, they will explore the algebraic strategies of substitution and elimination. At this stage, students will work with equations that have different slopes, thus guaranteeing a single solution, which they should be able to visualize by graphing. The two cases in which the two linear equations in a system have the same slope will not be explored until the final lesson. 14 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

15 1 Introducing the Lesson (5 min) Have students work in pairs to investigate the problem and graph the system of relations. Make materials such as grid paper, rulers, and graphing calculators available so that students can choose what they want to use. Then have groups of two or three pairs meet to discuss their ideas and strategies. Ask one pair of students to volunteer to explain their strategy to the class. Repeat this for a few strategies, such as using the x- and y-intercepts, using the y-intercept and slope, and using a graphing calculator. 2 Teaching and Learning (20 to 25 min) Investigate the Math Ask a student volunteer to read the problem and central question to the class. If you wish to scaffold the problem, making it easier for students to arrive at the equation 15x + 10y = 1200, you can ask students how much 100 kg of only almonds would be worth and how much 100 kg of only cranberries would be worth. This will provide hint questions that you can use with individual students. For example: How much will 100 kg of Matt s mixture be worth if he sells it for $12/kg? How much will the almonds in 100 kg of Matt s mixture be worth? Have students work in pairs to record their responses to the prompts in the investigation. Each partner may work on one of the relations in part A throughout the investigation. As students complete the investigation, ask questions to make sure that they understand the connections. Answers to Investigate the Math A. i) x + y = x + 10 y ii) = 12, 15x + 10y = 1200, or 0.15x y = B. The points on the line represent the amounts of almonds and cranberries that have a total mass of 100 kg. C. The points on the line represent the amounts of almonds and cranberries that have a value of $12/kg. D. The two lines intersect at (40, 60). The x-coordinate, 40, represents 40 kg of almonds. The y-coordinate, 60, represents 60 kg of cranberries. I can estimate quite accurately because the point is where grid lines meet on my graph. E. The point lies on both lines, so the coordinates represent the amounts of almonds and cranberries for which both equations are true. F = 100; 15(40) + 10(60) = = : Graphically Solving Linear Systems 15

16 Technology-Based Alternative Lesson This investigation can be done in much the same way with a graphing calculator as with grid paper and ruler. If TI-nspire calculators are available, students can use the Graphs & Geometry application as described in Appendix B-38 for the investigation and for the examples. They must rewrite each equation in the form y = mx + b to enter it. After graphing, direct students to look for the point of intersection as shown at the right for Example 3. Remind students that the intersection of the two lines must appear on the screen before they can use the Intersection Point(s) operation as described in Appendix B-47. The point, along with its coordinates, will appear on the screen. Answers to Reflecting G. There are two facts about almonds and cranberries that must be true: the total mass must be 100 kg, and the value must be $12/kg. Each of these facts needs a separate linear relation to represent it. H. When you graph both relations on the same axes, you can determine the point of intersection. Both relations are satisfied by this point, which means that both facts in the problem must be true. I. The coordinates satisfy both relations because they lie on both lines. 3 Consolidation (30 to 35 min) Apply the Math Using the Solved Examples Example 1 uses a different strategy for graphing each equation. Have students work in pairs, with each partner taking one equation and using the strategy that was used for the other equation in the Student Book; that is, using the x- and y-intercepts for y = 2x + 1 and the slope and y-intercept for x + 2y = 8. Then ask why the Student Book uses the strategies the other way round, and what this tells them about choosing strategies to graph linear equations. Example 2 requires students to use their modelling skills. Go through the steps for creating the second equation with the class, since these steps are more challenging than the steps for creating the first equations. Advise students to think carefully about what scales they will need on their axes. If they need a hint, ask: What are the intercepts of x + y = 450? Then have students work through the rest of the example in pairs. You might want to emphasize the importance of accuracy when graphing, as students should be able to see from the graph in the Student Book. Ask what students could do to be sure of their answer. Lead students to realize that they can check their answer by substituting the values into the original equations. 16 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

17 Example 3 presents a graphing-calculator solution to a problem that is similar to the problem in Example 2. Ask about advantages and disadvantages of this strategy, compared with the strategy in Example 2. Lead students to include the advantages of speed and accuracy with decimals, as well as the disadvantage of needing to convert an equation to the slope y-intercept form, which may be more difficult than the two-intercept strategy and may introduce errors. Answer to the Key Assessment Question If some students used a graphing calculator and others used grid paper to graph the system for question 13, have them compare graphs. 13. a) Let x represent the number of student tickets, and let y represent the number of non-student tickets. The total number of people equals 679, so x + y = 679. The revenue from students, 4x, plus the revenue from non-students, 7y, equals $3370, so 4x + 7y = b) Since the y-coordinate equals the number of non-students, 218 non-students attended the play. Closing When introducing question 16, suggest that students look through the examples if they need help. Discuss variations in the order, considering whether these variations are reasonable. If a student has a step out of order, explore why the student placed it in this position. The thinking is useful. Curious Math This Curious Math feature explores optical illusions created with parallel lines and intersecting lines. Prompt students to think about whether two lines that appear to be not quite parallel have a point of intersection. Ask: Does this suggest a test to determine if lines are parallel (even when they appear not be)? Answers to Curious Math 1. Answers may vary, e.g., this optical illusion, This optical illusion, the Müller Lyer illusion, the Wundt illusion, makes straight lines makes lines that are the same length appear to be curved. appear to be different lengths. 2. Answers may vary, e, g., 1.3: Graphically Solving Linear Systems 17

18 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students correctly create pairs of linear equations to model problems. Students choose appropriate strategies for graphing equations on a grid. Students use a graphing calculator to determine the point of intersection. Students use the coordinates of the point of intersection to answer the question correctly. If students misunderstand Students may have difficulty interpreting information in the problem, deciding what each variable should represent, or separating the information to determine how it is used for each equation. Students may have difficulty expressing equations in the slope y-intercept form, calculating intercepts, or graphing their results. Students cannot write each equation in the slope y-intercept form, enter the equations, use a thick line for one graph, or determine the point of intersection. Students may not understand what is represented by the point of intersection. They may have difficulty interpreting the coordinates correctly or writing a sentence to answer the question. Key Assessment Question 13 Students write two equations to represent the information correctly. Students correctly graph the equations they have written in part a). Students use the intersection point correctly to determine the number of non-students. Students are unsure which variable they should isolate. Students may incorrectly enter the equations in a graphing calculator, or they may incorrectly identify points on the lines if graphing by hand. Students may be unable to locate the intersection point using graphing technology, or they may misread a hand-drawn graph if it is not precisely drawn. An incorrect intersection point may be correct for the equations that students determined in part a). Differentiating Instruction How You Can Respond EXTRA SUPPORT 1. When choosing a strategy for graphing equations, students might find it helpful to ask themselves these questions: Do I want to use, or have I been asked to use, a graphing calculator? If not, is it better to use the x- and y-intercepts, or the slope and y-intercept? When answering both of these questions, students need to think about the features of the equation and consider the following questions: Does it already have y isolated on one side? If the equation involves fractions, would it help to multiply to create integers? 2. If students are having difficulty making the conceptual connection between the intersection of the two graphs and the solution to the linear system, suggest that they think about what a point on each line represents. This sets up the insight that the intersection point is on both lines, so it represents values that make both equations true, or values that satisfy both relations described in the problem. EXTRA CHALLENGE 1. Have students create problems that can be solved using a pair of linear equations and then exchange problems with a partner. 18 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

19 MID-CHAPTER REVIEW Big Ideas Covered So Far The solution to a linear system with two linear equations, where one or both have two variables, consists of the values that make both equations true. A pair of linear equations in x and y can be solved by graphing the equations and determining the point of intersection. Using the Frequently Asked Questions Have students keep their Student Books closed. Display the Frequently Asked Questions on a board. Have students discuss the questions and use the discussion to draw out what the class thinks are good answers. Then have students compare the class answers with the answers on Student Book pages 30 and 31. Students can refer to the answers to the Frequently Asked Questions as they work through the Practice Questions. Using the Mid-Chapter Review Ask students if they have any questions about any of the topics covered so far in the chapter. Review any topics that students would benefit from considering again. Assign Practice Questions for class work and for homework. To gain greater insight into students understanding of the material covered so far in the chapter, you may want to ask them questions such as the following: How is graphing a linear equation and determining the x-value for a given y-value similar to graphing two linear equations and determining the coordinates of the point of intersection? How are these two strategies different? When you graph two lines, what do you know about the coordinates of each point on one line? What do you know about the coordinates of each point on the other line? What do you know about the coordinates of the point that is on both lines? Why is this called the point of intersection? How do you decide whether to graph using grid paper or using a graphing calculator? How do you decide whether to determine both intercepts, or the slope and y-intercept? How might you use a table of values, rather than graphing, to determine the solution to a pair of equations, such as y = 3x + 2 and 2x y = 5? What are the advantages (if any) and disadvantages of this strategy, compared with graphing? Chapter 1 Mid-Chapter Review 19

20 1.4 SOLVING LINEAR SYSTEMS: SUBSTITUTION Lesson at a Glance GOAL Solve a system of linear equations using an algebraic strategy. Prerequisite Skills/Concepts Expand and simplify an algebraic expression. Solve a linear equation for one variable after substituting a known value for the other variable. Create a linear model to represent a situation. Graph a pair of linear equations and interpret the point of intersection. Specific Expectations Solve systems of two linear equations involving two variables, using the algebraic method of substitution [or elimination]. Solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method. Mathematical Process Focus Problem Solving Selecting Tools and Computational Strategies Connecting Student Book Pages Preparation and Planning Pacing 5 min Introduction min Teaching and Learning min Consolidation Materials grid paper ruler graphing calculator Recommended Practice Questions 4, 5, 7, 8, 10, 12, 15 Key Assessment Question Question 8 New Vocabulary substitution strategy Extra Practice Lesson 1.4 Extra Practice Nelson Website MATH BACKGROUND LESSON OVERVIEW In this lesson, students solve two linear equations in two variables using substitution, an algebraic technique that begins by determining a linear expression that involves one of the variables and is equal to the other variable. Students have encountered a simpler case of this general problem in Lesson 1.2, where one of the equations had the form y = constant. The process of solving two linear equations in two variables by substitution can be summarized as follows: Isolate a variable in one equation. Substitute for this variable in the other equation, and simplify. Solve the simplified equation to determine the value of the other variable. Substitute the value you determined into the first equation, and solve for the first variable. Check the values of both variables by substituting them into both of the original equations. (optional) The important business concept of a break-even point, where revenue, R, equals costs, C, is introduced. 20 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

21 1 Introducing the Lesson (5 min) Ask students to suggest two equations that are both in slope y-intercept form but have different slopes. Lead the class through the steps for graphing these equations on a displayed grid or on a graphing calculator screen. Ask the following questions: What problem can you create that could be represented by this graph? What is the solution to your problem? Point to one of the lines, and ask: What does each point on this graph represent for your problem? Repeat the question for the other line, and then ask: What does the point of intersection represent for your problem? Continue by asking students to suggest another problem for the same graph. 2 Teaching and Learning (10 to 15 min) Learn About the Math Example 1 introduces the technique of substitution for a system of equations in which the coefficients of both variables in one equation are 1. One strategy is to work through the example in class, asking volunteers to explain each step. Ask questions such as these: Why do we rearrange the first equation in this way? How does the substitution turn the second equation into an equation we can solve? Technology-Based Alternative Lesson If TI-nspire calculators are available, have students graph the relations to check the solutions they determined by substitution. Remind students to rewrite the equations in the form y = mx + b. Before using the Point(s) of Intersection operation, they need to ensure that the point of intersection is visible on the screen as described in Appendix B-38 and B-47. Answers to Reflecting A. Answers may vary, e.g., yes, because one equation might be easier to rearrange than the other to isolate a variable; no, because the equation you start with does not affect the point of intersection of the lines. B. She needed to substitute the value of n into one equation so that she could determine the value of m. C. You would solve for the variable n in the first equation and then substitute this expression for n into the second equation. You would solve for m and substitute the value of m into the first equation to determine the value of n. 1.4: Solving Linear Systems: Substitution 21

22 3 Consolidation (40 to 45 min) Apply the Math Using the Solved Examples The problem in Example 2 is slightly more complex than the problem in Example 1 because it involves decimals. However, the same strategy can be used to solve it. You might ask students to close their books and, in a guided whole-class setting, volunteer the steps in the solution. Then you could compare the class solution with the solution in the Student Book. Alternatively, you could ask students to explain each step in Wesley s thinking. Example 3 introduces the concept of the break-even point, an important real-world application. It is essential that students understand this concept, so that they understand why they need to create two equations in the same two variables. For Example 4, emphasize that there are two variables and two equations, giving four possible choices for a variable to isolate first. Have students work in pairs. Since the solution in the Student Book isolates y in equation 1, have each pair isolate either x in equation 1 or either variable in equation 2. Answer to the Key Assessment Question Students may find that sketching a diagram helps them visualize the situation for question 8. If necessary, remind students that the sum of the angles in a triangle is 180º. 8. The angles are 33, 44, and 103. Closing Have students read question 18. If students need help, discuss that the first step in solving this system is to write x = 8 4y. Students should focus on what happens in the next step, namely that the expression 8 4y is substituted for x in the equation 3x 16y = 3. Encourage students to think of other steps in which substitutions are made. Ask: What name would you give this strategy? 22 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

23 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students correctly create pairs of linear equations to model problems. Students correctly isolate for one variable and substitute the resulting expression into the other equation. Students understand that their expression for one variable in terms of the other variable represents the relationship between the coordinates of points on this line. If students misunderstand Students cannot correctly formulate pairs of linear equations to model problems. They may have difficulty deciding what information needs to be used for each equation or how the equations are linked. Students may apply the steps in the substitution strategy in an incorrect sequence and/or apply these steps incorrectly. They may make errors when determining an expression for one variable, substituting the expression into the other equation, or solving an equation. Students do not understand the significance of creating an expression for one variable in terms of the other variable. Key Assessment Question 8 Students write the correct equations for the sum and difference of the angle measures. Students correctly isolate for one variable and substitute the resulting expression. Students recall that the angles in a triangle add to 180 (Triangle Angle Sum Theorem) and determine the correct measure for the third angle. Students may use three variables instead of two, since three values are requested. Students may write one equation with one pair of variables and the other equation with a different pair of variables. Students may apply the steps in the substitution strategy in an incorrect sequence and/or apply these steps incorrectly. Students may determine the measure of only two of the three angles. Students may calculate an incorrect measure for the third angle because they recall the Triangle Angle Sum Theorem inaccurately (e.g., they may think that the angles add to 360 instead of 180 ). Differentiating Instruction How You Can Respond EXTRA SUPPORT 1. Since substitution is an algebraic strategy, students who learn visually may benefit from a visual strategy using algebra tiles, including both x tiles and y tiles. 2. If students have difficulty remembering or working through the steps in the substitution strategy, suggest that they create a flowchart for the strategy, using one of the examples as a guide. 3. A simplified version of the substitution strategy is to write both equations in slope y-intercept form and then equate the two expressions in x. Some struggling students may prefer this approach. After they have gained confidence, ask them if they really need to rearrange the second equation. Finally, ask them if they really need to begin by isolating y when it may be easier to isolate x. EXTRA CHALLENGE 1. If students are confident with algebra, have them work on the general system ax + by = e and cx + dy = f with a, b, c, d 0. (This system will be encountered in Lesson 1.6, Extending question 21, when students are challenged to solve it by elimination.) 2. Students can work in pairs to create systems of linear equations that can be solved by substitution. Ask them to write about their strategies for creating their systems. Students can then trade systems with other pairs to solve. 1.4: Solving Linear Systems: Substitution 23

24 1.5 EQUIVALENT LINEAR SYSTEMS GOAL Lesson at a Glance Compare solutions for equivalent systems of linear equations. Prerequisite Skills/Concepts Graph a linear relation using the x- and y-intercepts. Graph a linear relation using the slope and y-intercept. Expand and simplify an algebraic expression. Specific Expectation Solve systems of two linear equations involving two variables, using the algebraic method of [substitution or] elimination. Mathematical Process Focus Connecting Student Book Pages Preparation and Planning Pacing 5 10 min Introduction min Teaching and Learning 15 min Consolidation Materials grid paper ruler Recommended Practice Questions 5, 6, 9, 10, 12, 14, 16 Key Assessment Question Question 10 New Vocabulary equivalent systems of linear equations Extra Practice Lesson 1.5 Extra Practice Nelson Website MATH BACKGROUND LESSON OVERVIEW In this lesson, students connect the concept of equivalent systems of linear equations to the concept of generating new equations by adding, subtracting, and multiplying equations. Students encounter the possibility of using equivalent systems as a strategy for solving linear systems. This leads into Lesson Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

25 1 Introducing the Lesson (5 to 10 min) Reproduce the linear system and graph on page 41 of the Student Book so the whole class can see them, and ask students how to identify the solution. Then ask students to think about the system x + 2y = 10 and y = 6. What must the solution to this system be? Have a volunteer draw in the line y = 6. Ask: What must the solution to the system x + 2y = 10 and y = 3x be? Have a different volunteer draw in the line y = 3x. Ask: What is noticeable about all four lines? (They have a common intersection point.) What can you say about the solutions to the three different systems? (All the systems have the same solution.) 2 Teaching and Learning (35 to 40 min) Learn About the Math Example 1 establishes the concept of an equivalent system of linear equations. It also introduces the concept of adding or subtracting equations. Example 2 shows students that if each equation is multiplied by a number, the two new equations will form an equivalent system. Have students work in pairs, with each partner explaining one of the two examples to the other. In addition to the Reflecting questions, you might ask students to describe as many strategies as possible for producing new equations that form equivalent linear systems. If a hint is needed, ask students to consider combining the strategies used in the examples. Answers to Reflecting A. Yes. When you multiply an equation by a constant, the graph is identical so the point of intersection of the lines does not change. B. The graphs would have still been identical to the original graphs. C. The point of intersection is the solution to a system of linear equations. If the solution to the system is (a, b), then the system x = a and y = b represents the solution. The equation x = a represents a vertical line, and y = b represents a horizontal line. 1.5: Equivalent Linear Systems 25

26 3 Consolidation (15 min) Apply the Math Using the Solved Examples Example 3 makes the crucial point that combining by multiplying, adding, and subtracting can produce an equivalent linear system in which one of the new equations has only one variable. This makes solving the system significantly easier. Whether you choose to work through this example with the whole class or ask students to pair up, you should check that students understand the significance of this point. Answer to the Key Assessment Question After students complete question 10, arrange for them to share linear systems and explain their strategies for creating the systems and for verifying. Answers may vary, depending on decisions about creating the linear systems. 10. a) 3x 4y = 3 1 x y = 6 2 Add equations 1 and 2. 2x 5y = 9 3 Subtract equation 2 from equation 1. 4x 3y = 3 4 Equations 3 and 4 form a new system. Multiply equation 1 by 2. 6x 8y = 6 5 Multiply equation 2 by 1. x + y = 6 6 Equations 5 and 6 form another new system. b) Equations 1 and 5 can be rewritten as y = 0.75x Equations 2 and 6 can be rewritten as y = x 6. Equation 3 can be rewritten as y = 5 2 x 5 9. Equation 4 can be rewritten as y = 3 4 x + 1. Closing Graphing these equations shows the common intersection point, ( 3, 3). After students read question 16, ensure that they realize they should discuss the graphs, as well as the solutions of equivalent systems, in their answers. 26 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

27 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Given a system of equations, students use addition, subtraction, multiplication by a constant, or a combination of these operations to create an equivalent system. Students realize that equivalent systems have graphs with a common point of intersection, and that pairs of lines with a common point of intersection represent equivalent systems. Students recognize that if a linear system has the solution (a, b), the equations x = a and y = b form an equivalent system, which can be created by adding, subtracting, and/or multiplying the two given equations. If students misunderstand Students may make errors when adding, subtracting, or multiplying by a constant, or they may neglect to perform the operation on all terms in the equation(s). Students may not connect the point of intersection of a pair of lines with the solution to a linear system, or they may not connect a pair of lines to a system of linear equations. Students may not connect the equations x = a and y = b with lines, or they may have difficulty understanding how the original equations can be manipulated to yield x = a and y = b. Key Assessment Question 10 Students correctly form equivalent systems from the given system by adding, subtracting, and/or multiplying the given equations by a constant. Students use graphing technology or grid paper graphs to verify that the lines have a common point of intersection, or they solve the systems using algebraic techniques. They may also solve one system and then substitute the solution into the other systems to verify that all the equations are true. Students may not realize what operations will yield an equivalent system, or they may make errors when performing the operations. Students may not verify all three systems, or they may neglect to write all the equations in slope-intercept form first if using graphing technology. Differentiating Instruction How You Can Respond EXTRA SUPPORT 1. Visual learners may benefit from using graphing technology to verify that each operation performed on the equations does not change the point of intersection. 2. Students may become confused by the distinction between creating new equations and creating equivalent systems. Emphasize that any pair of equations, original or created from the original, can be used for an equivalent system. In fact, creating just one new equation from a given pair gives an additional two equivalent systems: the new equation plus either one of the original two equations. EXTRA CHALLENGE 1. Have students consider the family of equations y 5 = m(x + 2), where m can be any number, by asking the following questions: Which point do all the equations in this family go through? ( 2, 5) How can you use this family of equations to generate equivalent systems? (Choose different values of m to create equations for lines with a common point of intersection, ( 2, 5).) If you add two equations from this family, what happens? (The slope of the new equation is the mean of the slopes of the two equations added.) Are there any other lines that pass through ( 2, 5) but are not in this family? (x = 2) 1.5: Equivalent Linear Systems 27

28 1.6 SOLVING LINEAR SYSTEMS: ELIMINATION Lesson at a Glance GOAL Solve a linear system of equations using equivalent equations to remove a variable. Prerequisite Skills/Concepts Expand and simplify an algebraic expression. Solve a linear equation for one variable after substituting a known value for the other variable. Create a linear model to represent a situation. Understand the concept of equivalent linear systems. Specific Expectations Solve systems of two linear equations involving two variables, using the algebraic method of [substitution or] elimination. Solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method. Mathematical Process Focus Problem Solving Selecting Tools and Computational Strategies Connecting Student Book Pages Preparation and Planning Pacing 5 min Introduction min Teaching and Learning min Consolidation Recommended Practice Questions 6, 8, 10, 11, 12, 14, 16, 18 Key Assessment Question Question 16 New Vocabulary elimination strategy Extra Practice Lesson 1.6 Extra Practice Nelson Website MATH BACKGROUND LESSON OVERVIEW In this lesson, students use their understanding of equivalent systems from Lesson 1.5 to understand and apply the elimination strategy for solving linear systems. Solving by elimination provides an alternative to solving by substitution, which they learned in Lesson 1.5. The process of solving two linear equations in two variables by elimination can be summarized as follows: Multiply each equation by a constant, if necessary, to yield an equivalent system in which the coefficients of one variable are either the same or opposite. Create an equation in one variable by subtracting or adding the equations. Isolate the remaining variable. Substitute the value of this variable into one of the original equations, and solve for the other variable. 28 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

29 1 Introducing the Lesson (5 min) Begin by having the class consider this system: 2x + y = x y = 31 2 Ask students to predict what will happen to y when the two equations are added. (It will be eliminated.) From there, some students will be able to realize how to solve for x and then for y. Ask students whether they think this strategy has any advantages over the graphing or substitution strategies they have already encountered. 2 Teaching and Learning (25 to 30 min) Learn About the Math Example 1 presents a problem that can be efficiently solved using an elimination strategy. The two solutions highlight the idea that either of the variables can be eliminated. Have students work through the example in pairs, with each partner explaining one of the solutions to the other. Then have the pairs respond to the Reflecting prompts. Bring the whole class together to discuss their answers. Answers to Reflecting A. They both subtracted or added multiples of the original equations, chosen so that the coefficients of one of the variables would result in 0 in the resulting single equation. B. The original equations did not have matching or opposite coefficients for either variable. C. Add when the coefficients of one of the variables are opposite. Subtract when the coefficients are equal. D. Answers may vary, e.g., Leif s strategy because it involves addition. With addition, there is less chance of a calculation error, such as leaving a double negative as a negative. 1.6: Solving Linear Systems: Elimination 29

30 3 Consolidation (25 to 30 min) Apply the Math Using the Solved Examples Use Example 2 to highlight why carefully choosing the variable to eliminate can make a linear system easier to solve. For this system, y is easier to eliminate than x because its coefficients are common factors of a relatively small number, making the calculations manageable. Ask students to read through the example and then answer these questions: Would it have been easier to eliminate x rather than y? Why or why not? Also ask students to explain why it is important to verify the solution by substituting the values back into the original equations. Example 3 applies an elimination strategy to a real-world problem. Work through the example with the class, but eliminate y instead of x. (This leads to 1 the equations x = 80 and x = 160.) Ask students to suggest reasons why x 2 is the better variable to eliminate. Answer to the Key Assessment Question The following reasoning may help students who need support for question 16. Let x represent the number of students who ordered chicken, and let y represent the number of students who ordered pasta. There are 240 students, so x + y = 240. Chicken dinners are $12 each, so the cost of the chicken dinners for x students is 12x. The cost of the pasta dinners is 8y. The total cost is $2100, so 12x + 8y = After students have completed the question, arrange for them to explain how they solved the equations to classmates, comparing differences in strategies. 16. a) The students ordered 45 chicken dinners. b) The students ordered 195 pasta dinners. Closing Have students read and answer question 18. Their explanations should mention an equivalent system with either x-coefficients or y-coefficients equal or opposite. As well, students could compare the number of calculation steps in substitution and elimination, and they could mention whether fractions appear before the final step of the solution and whether a graphing strategy is likely to be accurate. 30 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

31 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students correctly create pairs of linear equations to model problems. Students correctly identify and apply elimination strategies to solve problems. If students misunderstand Students cannot use variables to represent information, or they cannot use given information to write equations. Students do not recognize when to multiply, add, and/or subtract equations, or they make calculation errors when combining or solving equations. Key Assessment Question 16 Students write two equations to represent the information given in the problem. Students correctly identify and apply an elimination strategy (such as multiplying the equation x + y = 240 by 8 and subtracting from the equation 12c + 8p = 2100). Students cannot write two equations to represent the information given in the problem, or they write incorrect equations. Students may not be able to select an appropriate strategy to eliminate a variable, or they may calculate incorrectly. Differentiating Instruction How You Can Respond EXTRA SUPPORT 1. Encourage students who have difficulty using elimination to adopt this simplified approach: Choose either variable. Multiply each equation by the coefficient of this variable in the other equation. Subtract one of the new equations from the other or add the equations; one variable will be eliminated. As students gain confidence, they should develop their own approach for deciding which variable they should initially choose. 2. Students can work in pairs, with one student explaining the steps for the other to record. Encourage students to help their partner with explaining or recording, as needed. Have students switch roles for different problems. EXTRA CHALLENGE 1. Challenge students to create linear systems to solve using elimination. Students can work on their own or in pairs and solve each other s systems. 2. Students can create a graphic organizer of their choice to compare solving linear systems with solving by substitution and solving by elimination. Encourage creativity. Students can present their graphic organizers to classmates and explain their thinking. 1.6: Solving Linear Systems: Elimination 31

32 1.7 EXPLORING LINEAR SYSTEMS Lesson at a Glance GOAL Connect the number of solutions to a linear system with its equations and graphs. Prerequisite Skills/Concepts Solve a system of linear equations by substitution. Solve a system of linear equations by elimination. Graph a linear system. Specific Expectations Solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination. Solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method. Mathematical Process Focus Reasoning and Proving Connecting Representing Student Book Pages Preparation and Planning Pacing 5 10 min Introduction min Teaching and Learning 20 min Consolidation Materials graphing calculator, or grid paper and ruler Recommended Practice Questions 2, 3, 4, 5, 6 Nelson Website MATH BACKGROUND LESSON OVERVIEW So far, students have been only working with pairs of equations whose graphs have different slopes. When the two equations in a linear system have different slopes, the linear system always has a solution. When the slopes are the same, the ratio of the x-coefficients is the same as the ratio of the y-coefficients. The graphs may be either parallel and identical (infinitely many solutions) or parallel and distinct (no solutions). Students informally develop the following ratio test for the number of solutions to the system ax + by = e and cx + dy = f: If a : c b : d, the system has one solution. If a : c = b : d e : f, the system has no solutions. If a : c = b : d = e : f, the system has infinitely many solutions. 32 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

33 1 Introducing the Lesson (5 to 10 min) Have students work in small groups. Ask each group to discuss important points for solving a linear system by substitution and by elimination. After a few minutes, ask each group to record what they think is the most important point for each strategy. Invite a group to read one of its points. Continue for other groups, until each group has read a point. Some groups may want to read more than one point. 2 Teaching and Learning (30 to 35 min) Explore the Math Have students work in pairs to record their responses to the prompts in this exploration. There are several ways that pairs can divide up the work. For instance, one partner can do the algebra and the other can do the graphing. Or the partners can solve and graph system A together and then each take one of the remaining systems. However, they should discuss their findings for parts D through G together. For part D, you may need to clarify the distinction between systems B and C for students, since they may view system C as having no solution rather than having many. For parts E through G, some students may find it helpful to put the equations into slope y-intercept form. In this way, they should be better able to connect the slope to the pattern of the coefficients in the equation. Answers to Explore the Math A. System A: 2x + 3y = 4 1 4x 3y = 1 2 Add equations 1 and 2. 2x = 5 x = 2.5 Substitute for x in equation 1. 2(2.5) + 3y = y = 4 3y = 9 y = 3 (2.5, 3); one solution 1.7: Exploring Linear Systems 33

34 System B: 3x + 2y = 6 1 6x + 4y = 5 2 Multiply equation 1 by 2. 6x + 4y = 12 3 Subtract equation 3 from equation 2. 0 = 7 no solutions System C: x y = 5 1 3x 3y = 15 2 Multiply equation 1 by 3. 3x 3y = 15 3 Subtract equation 3 from equation 2. 0 = 0 Any point on x y = 5 is a solution; infinitely many solutions. B. The graphs intersect at one point because there is one solution. C. System B: The graphs do not intersect because there are no solutions. System C: The graphs are identical because any pair of coordinates, (x, y), is a solution of x y = Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

35 D. System A: The lines have different slopes, and (2.5, 3) is the point of intersection and the one algebraic solution. System B: There is no graphical solution because the lines are parallel and do not intersect. There is no algebraic solution because 0 = 7 is not true. System C: The lines are identical, so any point on the lines is a graphical solution. The algebraic solution is any (x, y) with x y = 5, because 0 = 0 is always true. E. Answers may vary, e.g., In system A, the slopes of the lines are different so there will be one point of intersection. In system B, the slopes of the lines are the same but the lines have different y-intercepts, so they do not intersect. In system C, the lines are the same because the slopes and y-intercepts are the same. The lines intersect at every point. F. No. The solution to a system is a point of intersection for the graphs. When the lines have different slopes, they intersect once. When they have the same slope, they either do not intersect at all or they intersect at every point. G. Case 1: A linear system has one solution; the equations have different slopes; the lines are not parallel and intersect at a single point. Case 2: A linear system has no solution; the equations have the same slope but neither equation is a multiple of the other; the lines are parallel and do not intersect. Case 3: A linear system has infinitely many solutions; the equations have the same slope, and one is a multiple of the other; the lines are identical. Answers to Reflecting H. The values of A are in the same ratio as the values of B, but not the values of C. The lines are parallel and not the same. I. The values of A, the values of B, and the values of C are all in the same ratio. The lines are identical. K. The values of A are not in the same ratio as the values of B. 3 Consolidation (20 min) Students should be able to create or identify systems of each of the three types: no solution, one solution, and infinitely many solutions. Students should understand and be able to explain the connections among linear systems, their graphs, and the number of solutions they have. 1.7: Exploring Linear Systems 35

36 CHAPTER REVIEW Big Ideas Covered So Far The solution to a linear system with two linear equations, where one or both have two variables, consists of the values that make both equations true. A pair of linear equations in x and y can be solved by graphing the equations and determining the point of intersection. A pair of linear equations in x and y can be solved by substitution: solving one equation for one variable and substituting this expression into the other equation. Different linear systems can be equivalent; that is, they can have the same solution. A pair of linear equations in x and y can be solved by elimination: creating an equivalent system with either the x-coefficients or the y-coefficients the same or opposite. A pair of linear equations in x and y has no solutions when the graphs of the equations are parallel and not identical, one solution when the graphs are not parallel, or infinitely many solutions when the graphs are identical. Using the Frequently Asked Questions Have students keep their Student Books closed. Display the Frequently Asked Questions on a board. Have students discuss the questions and use the discussion to draw out what the class thinks are good answers. Then have students compare the class answers with the answers on Student Book pages 60 and 61. Students can refer to the answers to the Frequently Asked Questions as they work through the Practice Questions. Using the Chapter Review Ask students if they have any questions about any of the topics covered so far in the chapter. Review any topics that students would benefit from considering again. Assign Practice Questions for class work and for homework. To gain greater insight into students' understanding of the material covered so far in the chapter, you may want to ask questions such as the following: If it is easy to isolate one variable in one of the given equations, which algebraic strategy might you use? Would a different strategy work? Why? Which algebraic strategy would you use if one variable has equal or opposite coefficients in the given equations? Suppose that a linear system has the solution (2, 5). Write an equivalent system, with graphs that are a horizontal line and a vertical line. What is an example of a linear system with no solutions? How do you know that it has no solutions? What is an example of a linear system with one solution? How do you know that it has one solution? What is an example of a linear system with infinitely many solutions? How do you know that it has infinitely many solutions? 36 Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

37 CHAPTER 1 TEST For further assessment items, please use Nelson's Computerized Assessment Bank. 1. Art is buying games for his Z-Cube. He has $120 to spend and wants to buy some single games at $8 each and some combo packs at $24 each. a) Create a table of values for Art s choices. b) Draw a graph to represent Art s choices. 2. Erin is playing a board game in which she can buy apartments at $300 each and houses at $500 each. She has a total of $3500 to spend. Which equation could model Erin s options? a) 3x + 35 = 5y c) y = 3x b) 5x + 3y = 35 d) 500x 300y = Solve the equation 7x 5y = 150 if x = Use a graphing calculator to solve the equation y = x, if y = Graph the equations y + x = 32 and x 2y = 23 to determine their solution. 6. Yannick sets out on a road trip. He plans to travel a total distance of 375 km. For the first part of the trip, he travels at 100 km/h. For the second and final part of the trip, he travels at 75 km/h. If Yannick s total travel time is 4 h 15 min, how far does he travel in the second part of the trip? 7. Solve each system of equations by elimination. a) 10x 4y = 12 b) 3x + 5y = 7 x + y = 32 2y x = 5 8. Derek is starting up a craft business, selling metal figures. He needs to buy $4500 of metalworking equipment, as well as the raw materials for the figures, which cost $13 per figure. He plans to sell each figure for $88. How many figures does Derek need to sell to break even? a) 75 b) 52 c) 60 d) Keisha likes to have yogurt and raspberries for breakfast. She wants her breakfast to have 25 g of carbohydrates and 10 g of protein. She finds these nutritional values online: 100 g of yogurt contains 7 g of carbohydrates and 5 g of protein. 100 g of raspberries contains 12 g of carbohydrates and 1 g of protein. How much of each, to the nearest 10 g, should Keisha eat? 10. Which system of linear equations is equivalent to the system 3x + 2y = 4 and 5y 2x = 9? a) 3x + 2y = 4 and x = 1 b) x + y = 2 and 5y 2x = 9 c) 3x + 2y = 4 and y = x 3 d) x + y = 1 and x y = 2.5 Chapter 1 Test 37

38 11. Suppose that you are given three linear equations, each with a different slope. The linear system of equations 1 and 2 is equivalent to the linear system of equations 1 and 3. a) What must be true about the graphs of these equations? b) What can you say about the linear system of equations 2 and 3? Explain. c) What can you say about the solution to equations 2 and 3? 12. Use elimination to solve the linear system 2x + 5y = 4 and 3x + 7y = A group of students organized a canoe trip in Algonquin Provincial Park. They travelled at 8 km/h while canoeing but only 2 km/h while portaging between waterways. On the third day of the trip, the group travelled 36 km in 6 h. How far did they portage? 14. Without solving, predict the number of solutions to each system of equations. a) 14x + 3y = 12 14x 3y = 0 b) y = 3x 2 3x + y + 2 = 0 c) 2x 3y = 4 6y = 4x + 15 d) 3x 6y = y = x Principles of Mathematics 10: Chapter 1: Systems of Linear Equations

39 CHAPTER 1 TEST ANSWERS 1. a) Answers may vary, e.g., let x represent the number of single games, and let y represent the number of combo packs. x b) y b) 3. y = (29, 3) km 7. a) (10, 22) b) ( 1, 2) 8. c) 9. Let x represent the amount of yogurt (in 100 g) and let y represent the amount of berries (in 100 g). Then the system is 7x + 12y = 25, 5x + y = c) The solution is about 180 g of yogurt and 100 g of raspberries. 11. a) The graphs of the equations all intersect at the same point. b) It is equivalent to both of the other linear systems, because all three equations intersect at a common point. c) It is the same as the solution to the other two linear systems, at the coordinates of the common intersection point , km 14. a) one solution b) one solution c) no solutions d) infinitely many solutions Chapter 1 Test Answers 39