unit 6: systems of linear equations (3...

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Unit 6 Plan CT Algebra I Model Curriculum Version 3.0

Unit 6: Systems of Linear Equations (3 Weeks)

UNIT OVERVIEW

Storyline In previous units, students studied linear functions and used a linear function to investigate the relationship between two variables. In this unit, students will represent, compare and analyze two linear equations, look for common solutions and use this information to make choices between competing situations in real world contexts. Students will solve systems of equations numerically, graphically, and algebraically. They will be able to explain what the solution of a system of linear equations represents in the context of various applications such as those used by business leaders, economists, scientists, engineers, nutritionists, racecar drivers, and athletes. They also will explore the special cases of parallel lines (no solution) and identical lines (infinite solutions). In the first investigation, students may work in small groups to determine whether or not women’s salaries within a specific salary range will ever equal the men’s salaries. Students will use their knowledge from Unit 5 to calculate the intersection point of two linear functions using the graphing calculator. Once they have the point of intersection, they will explain what the point of intersection means in the context of the problem. The next application will have the students explore under what conditions one gym membership is more economical than another. Students may solve the problem by working in small groups using different approaches such as making a table, solving an equation, graphing by hand, and graphing on the calculator. Then they share their solutions, with the class. The second investigation uses a non-profit organization as a context to explore solving systems of linear equations by substitution. Through questions posed by the teacher, the students will be guided through the process of how to solve a system of equations by substitution. This strategy builds upon students’ skill evaluating expressions given the value of one or more variables. In order to explore the case when two equations are given in slope-intercept form, the students will study car racing where the slower car receives a head start. They also will study another application, the economics of the breakeven point, a situation in which revenue equals cost. In the final investigation students work with linear equations that model situations such as a computer assembly line and designing a fund raiser. These scenarios are not easily solved using the substitution method and therefore motivate the need for solving systems of equations using elimination. Students will use and explain the algebraic principles that support the elimination method. Through the three investigations in this unit, students will understand how to solve equations involving two unknowns, both algebraically and graphically. Students will identify the point of intersection of the two lines as the solution of the system of equations and then interpret the solution in the context of the problem. Students will recognize when one method of solving a system of linear equations is more advantageous than another.

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Essential Questions • What does the number of solutions (none, one or infinite) of a system of linear equations

represent? • What are the advantages and disadvantages of solving a system of linear equations

graphically versus algebraically? Enduring Understandings

• A system of linear equations is an algebraic way to compare two equations that model a situation and find the breakeven point or choose the most efficient or economical plan.

Unit Contents Investigation 1: Solving Systems of Linear Equations (3 days) Investigation 2: Solving Systems of Linear Equations Using Substitution (2 days) Investigation 3: Solving Systems of Linear Equations Using Elimination (3 days) Performance Task: Community Park (4 days) End of Unit Test (2 days, including review)

Common Core Standards Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning. Practices in bold are to be emphasized in the unit.

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Standards Overview

• Create equations that describe numbers or relationships • Solve systems of equations • Represent and solve equations and inequalities graphically

Standards with Priority Standards in Bold A-CED 3. Represent constraints by equations or inequalities, and by systems of equations and/or

inequalities, and interpret solutions as viable or nonviable options in a modeling context. A-REI 5. Prove that, given a system of two equations in two variables, replacing one equation by

the sum of that equation and a multiple of the other produces a system with the same solutions.

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A-REI 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A-REI 11. Explain why the x-coordinates of the points where the graphs of the equations y

= f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear functions.*

Vocabulary Addition Property of Equality Breakeven Point Elimination Method for Solving Systems of

Equations Fixed Cost Multiplication Property of Equality Profit Revenue

Solution of a System of Linear Equations Substitution Method for Solving Systems Substitution Property of Equality System of Linear Equations Total Cost Transitive Property of Equality Variable Cost

Assessment Strategies

Performance Task: Community Park Students will complete a plan for a community park that contains a basketball court, walkways, and a feature (like a fountain or gazebo) at the intersection of the walkways. Other Evidence (Formative and Summative Assessments)

• Exit slips • Class work • Homework assignments • Journal entries • Unit 6 Test

Unit 6 Materials List

Investigation 1 Graph paper Graphing Calculators Investigation 2 Graphing Calculators Investigation 3 None Performance Task Graph paper

CT Algebra I Model Curriculum Web Sites for Unit 6

Where used Web site address Last checked

Investigation 1

U.S, Census Bureau http://usgovinfo.about.com/od/censusandstatistics/Census_and_Statistics.htm .

11/19/12

Investigation 2

Heifer Project www.heifereducation.org www.Heifer.org

11/19/12

Investigation 2 Drag Racing http://www.dragtimes.com/drag-racing-videos.php http://uhaweb.hartford.edu/rdecker/

11/19/12

Investigation 3 Computer Production http://www.thefutureschannel.com/hands- on_math/computer_problems.php

11/19/12

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Unit 6 – Investigation 1 Overview CT Algebra I Model Curriculum Version 3.0

Unit 6: Investigation 1 (3 Days)

SOLVING SYSTEMS OF LINEAR EQUATIONS CCSS: A-REI #6, A-REI #11 Overview In this investigation, students will solve systems of linear equations by making tables, solving linear equations in one variable, and graphing lines (both by hand and with the graphing calculator). They will find and interpret solutions of systems of linear equations and use systems of linear equations to solve real world problems. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? • Students will write equations to model a situation, graph equations (both by

hand and using the graphing calculator), find the point of intersection and interpret the solution in the context of the problem.

• Students will solve a system of linear equations that represents a real world situation graphically and numerically.

• Students will explain what the solution to a system of linear equations means in the context of the problem.

Assessment Strategies: How Will They Show What They Know? • Exit Slip 6.1 asks students to solve a system of linear equations graphically

and to explain what the solution of that system represents. • Journal Entry asks students to explain in their own words how to find the

solution of a system of linear equations graphically. Launch Notes The gender wage-gap is a real-life context which can be used to engage students in the first activity. Historically, women in the labor force have earned less than men. Salaries in the range $50,000 to $74,999 have traditionally been considered “middle” to “upper middle” class. Not surprisingly, the percent of men in this group is greater than the percent of women. However, this may change in the future. Display the data table on an overhead projector or digital board and begin the lesson by asking the question: “Will the percentage of women earning $50,000 - $74,999 per year ever be equal to or surpass the percentage of men earning $50,000 - $74,999 per year?” Inform students that they will use a system of linear equations to solve this problem and that they will discover throughout this unit that systems of linear equations can be used to solve problems, make decisions, and make predictions.

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Closure Notes The investigation culminates with students successfully demonstrating their ability to solve a system of linear equations graphically. This may be done with Exit Slip 6.1. Immediately before or after administering the exit slip, have students discuss the meaning of the solution of a system of equations. They should recognize that the point of intersection on the graph represents an ordered pair that satisfies both equations. Teaching Strategies

Differentiated Instruction (For Learners Needing More Help) You may start by giving students two equations that are easy to plot. Ask them to graph the system by hand and locate the point of intersection. Then move into the example that arises from the real world situation.

I. Begin the class by displaying the data table below and asking the class, “Will the

percentage of women earning $50,000 - $74,999 per year ever be equal to or surpass the percentage of men earning $50,000 - $74,999 per year?” (See Activity 6.1.1a Will Women Catch the Men?). Let the students think about the information for a few minutes and offer any comments.

Number of years

since 2000

% of men earning

$50,000 – $74,999

% of women earning

$50,000 – $74,999

2 20.1 13.0 3 20.2 13.3 4 20.5 14.2 5 20.7 15.1

The students may be able to predict that the women within the income bracket specified will indeed ‘catch’ the men since the percentage of women earning $50,000 - $74,999 is increasing at a more rapid rate then the percentage of men. If so, then ask them, “If these trends continue, when will the women catch the men and, at that moment, what percentage of women will be earning $50,000 - $74,999?”

If students do not think about using scatter plots and trend lines to represent the data, then probe them by asking, “What strategies have you learned this year that may be useful in this situation?” and/or “What do you notice about the data?”

Students may choose to make a scatter plot and fit trend lines by hand or they may choose to use calculators to find the least squares regression lines. Students should notice that the slope of the linear model for the women is greater than the

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slope of the linear model for the men, indicating that, if the rates stay constant, eventually the women will catch up. Ask them “How will you use these equations to answer the question: If these trends continue, when will the women catch the men and what percentage of women will be earning $50,000–$74,999?”

Students should realize that by extending the trend lines, they can find the point of intersection, and answer the question. If they are using a calculator they may trace to the point of intersection or use the intersect feature. To do so they select 2nd Trace (Calculate), 5: intersect, and choose their two lines as the 1st curve and 2nd curve. The point of intersection is (16.20, 23.04). You should then help them interpret the meaning of this point. The women should catch up to the men in the year 2016 at which time approximately 23% of each gender will be earning between $50,000 and $74,999. In the course of the discussion you should revisit important ideas from Unit 5, including the meaning of the correlation coefficient—in both cases r ≈.98—and the difference between interpolation and extrapolation. Students may suggest that the recession of 2008 may affect the trends and lessen the accuracy of the prediction. You may ask students to help develop the definition of a system of linear equations and what is meant by a solution of a system of equations. As they solve more problems in this unit, students may refine their definitions. Note that Activity 6.1.1a Will the Women Catch the Men is designed for an open-ended approach to this problem. You may choose to use Activity 6.1.1b Will the Women Catch the Men which is more structured version of the activity if you wish to have students work in groups after the problem has been presented to the entire class.

Differentiated Instruction (For Learners Needing More Help)

This is an opportunity to build on the skills learned in Unit 5 by calculating two linear regression equations and finding the point of intersection. Allow students to use the formula reference section of their notebook, which could include a procedure card that shows how to create a scatter plot on the graphing calculator, calculate a linear regression, and calculate a point of intersection.

Technology Tips: If students use technology to find regression lines, guide students to see that solving this problem using technology utilizes skills that they learned in Unit 5. Note that only three lists are needed instead of four – the independent variable is the same for both dependent variables, thus it only needs to be entered once. Therefore, the students could input the data in the table into Lists 1, 2, and 3. Then, they need to use Lists 1 and 2 (number of years since 2000

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and men’s data) to graph scatter plot 1. Next, they should use the data in Lists 1 and 3 (number of years since 2000 and women’s data) to graph scatter plot 2. Note that when calculating the linear regression for the men, they can use the command LinReg(ax+b) L1, L2. The equation they should get is ! = .21! + 19.64. When calculating the linear regression equation for women they can use the command LinReg(ax+b) L1, L3. The equation they should get is ! = .72! + 11.38. They may then enter the equations in Y1 and Y2 in the “Y =” menu. Some newer versions of the TI-83 family of calculators allow you to do this in one step via the commands LinReg(ax+b) L1, L2, Y1 and LinReg(ax+b) L1, L3, Y2. Be sure to ask students to explain the meaning of the slopes of the two lines. Also, when they find the point of intersection, have them substitute the coordinates into both equations to show that both equations are satisfied at the point of intersection. Students may get an error message on their calculator depending on what they chose for their window settings. Remind them that they must be able to see the point of intersection in order to calculate it. Thus, they must modify their window.

Differentiated Instruction (Enrichment) If students are interested in exploring data for earnings between $75,000 and $99,999 a year, they may use the table below. You may choose to use these data as an extension to the lesson as well.

Linear regression equation for men: y = 0.2x + 7.4667 Linear regression equation for women: y = 0.45x + 2.3667 Intersection Point: (20.4, 11.55) Another possible extension would be for students to go the U.S. Census Bureau website and research the earnings for other years and/or income levels. Source: http://usgovinfo.about.com/od/censusandstatistics/Census_and_Statistics.htm

Number of

years since 2000

% of men earning

$75,000 - $99,999

% of women earning

$75,000 - $99,999

3 8.1 3.7 4 8.2 4.2 5 8.5 4.6

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II. Activity 6.1.2 Choosing a Gym provides students an opportunity to solve real world problems using multiple approaches. As students work on problem 1, circulate to see the approaches that students are using to solve the problem. Students may solve the problem by making a table, using a guess and check approach, solving the equation 10! + 120 = 20!, graphing by hand, or graphing on the graphing calculator. If they have answered the question by using one of these methods, then challenge them to see if they can represent the situation in a different manner or solve the problem using a different method. You may have different pairs of students present their solutions to the class so that solving the problem graphically, algebraically, and via a table are all presented. Discuss the pros and cons of each method. Discuss the answer to part (d) and ask the class what it means to be a solution of a system of linear equations.

Problems 2 and 3 in Activity 6.1.2 present similar scenarios. They introduce students to two special situations. In problem 2 the two equations are represented by the same line and there are an infinite number of solutions. In problem 3 the two lines are parallel so there is no point of intersection. Ellie’s gym is less expensive no matter how many months Natasha uses the gym.

Group Activity Structure the assignment of Activity 6.1.2 to be given to pairs of students. For example, one member may do question 1a while the other is working on 1b. They can then compare answers and work together on 1c and 1d. For question 2b, each student can be responsible for writing one of the equations, then checking the other’s work. Have students take joint responsibility for the final question in each set (1d, 2e, and 3e) in which they must explain the rationale for their choices and ask them to be prepared to share their reasoning with the class.

III. Activity 6.1.3 Solving Systems of Equations by Graphing provides students

additional practice in using the graphing method to solve systems of equations. Students may need to be reminded of the different techniques they learned in Unit 4 for graphing the equation of a line. They may pick two or more points, or start with the y-intercept and use the slope to find a second point. This selection of systems includes points of intersection in all four quadrants and equations with fractional coefficients. Re-emphasize that a solution to a system of equations consists of an ordered pair (x, y) and that they should check to be sure the solution satisfies both equations.

In problems 6 and 7 students again encounter situations where there is an infinite number of solutions and where there is no solution. Dependent and inconsistent systems will be studied in more detail in the next investigation. Finally, in Problem 8, the intersection point lies outside of the graph given. Students will need to reason that when the slopes are different the lines will eventually intersect.

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They may use another sheet of graph paper or a calculator to find the solution, which is (6, 23).

Up until this point, all the equations have been given in slope-intercept form. Activity 6.1.4 Systems with Equations in Different Forms extends the graphing method to systems that contain lines in standard form or slope-intercept form. Have students graph these equations in two ways: (1) by finding two or more points to make a graph by hand and (2) by solving for y in terms of x so that the equation is in slope-intercept form and can be entered into the Y= menu on the calculator.

Have students complete Exit Slip 6.1, which requires them to graph a system of linear equations graphically and explain what the solution represents.

Journal Entry Explain, in your own words, how to find the solution of a system of linear equations graphically.

Resources and Materials

• Activity 6.1.1a Will the Women Catch the Men • Activity 6.1.1b Will the Women Catch the Men • Activity 6.1.2 Choosing a Gym • Activity 6.1.3 Solving Systems of Equations by Graphing • Activity 6.1.4 Systems with Equations in Different Forms • Exit Slip 6.1 Solving a System by Graphing • Graph paper • Calculators • Bulletin board for key concepts

Name: Date: Page 1 of 1

Activity 6.1.1a CT Algebra I Model Curriculum Version 3.0

Will the Women Catch the Men?

Income Bracket Earnings by Gender (2002 – 2005)

Number of years since 2000

% of men earning

$50,000 - $74,999

% of women earning

$50,000 - $74,999

2 20.1 13.0

3 20.2 13.3

4 20.5 14.2

5 20.7 15.1

Source: U.S. Census Bureau Based on the data given, will the percent of women earning $50,000 to $74,999 ever catch up to the percent of men earning $50,000 to $74,999? And if it will, when? In the space below, write notes based on the class discussion.


Activity 6.1.1b CT Algebra I Model Curriculum Version 3.0

Will the Women Catch the Men?

Income Bracket Earnings by Gender (2002 – 2005)

Number of years since 2000

% of men earning

$50,000 - $74,999

% of women earning

$50,000 - $74,999

2 20.1 13.0

3 20.2 13.3

4 20.5 14.2

5 20.7 15.1

Analyzing the Data 1. What is the independent variable? 2. What is the dependent variable? 3. Draw scatter plots for both the men

and the women on the same set of axes. (You may use a calculator, the graph at the right, or another sheet of graph paper.)

4. Fit a trend line for the men. 5. Fit a trend line for the women. 6. Find the point of intersection for the two trend lines. 7. Interpret the meaning of the point of intersection. 8. Make a prediction: will the percent of women ever equal the percent of men earning $50,000-

$74,999. If so, when?


Activity 6.1.2 CT Algebra I Model Curriculum Version 3.0

Choosing a Gym

1. There are two gyms in Groton, CT advertising promotional plans. The membership plans are as follows:

Gym A: $120 one time membership fee plus $10 per month

Gym B: No membership fee and $20 per month

a. Mark’s father is stationed at the Naval Submarine Base for the next nine months and his

family lives in Groton. Which gym should Mark join during their nine months in Groton? Why?

b. José is planning to attend the University of Connecticut at Avery Point to earn his Bachelor’s Degree in Coastal Studies. José will live in off-campus housing in Groton and he is trying to decide which of the two gyms to join during his four years of college. Which gym should José join? Why?

c. For how many months would it be best to join Gym A? When would it be best to join Gym B? Explain.

d. When would it not matter which gym a person joined? Explain how you made your decision.



2. Susan is trying to choose between two gyms. Susan loves Zumba (the latest cardio/aerobic craze) and must decide between two gyms that both offer Zumba. The gyms in her neighborhood offer the following membership plans:

Phoenix Gym: $40 one time membership fee, $10 one time towel fee, and $25 per month Rocker Spa: $50 one time membership fee, $15 per month for general membership, and an additional $10 per month for Zumba classes

a. If you had to choose a gym without doing any math, which would you choose? Why?

b. Write the equations for both gyms. Think about the meaning of each slope in context of the problem.

Phoenix Gym Rocker Spa

c. Solve the system by graphing.

d. What is the solution?

e. Which Gym should Susan join? Why?



3. Natasha is also trying to decide between two gyms, but her passion is kickboxing. Should she join Ellie’s Gym, a typical gym with kickboxing classes, or The Kick Box, a kickboxing gym, if she only wants to attend kickboxing classes and wants the best deal?

Ellie’s Gym: $60 one time membership fee and $10 per month for general membership and an additional $10 per month for kickboxing classes.

The Kick Box: $75 one time membership fee and $20 per month for kickboxing classes. a. If you had to choose a gym without doing any math, which would you choose? Why? (no

wrong answers)

b. Write the equations for both gyms. Think about the meaning of each slope in context of the problem.

Ellie’s Gym The Kick Box

c. Solve the system by graphing.

d. What is the solution?

e. Which gym should Natasha join? Why?



Solving Systems of Equations by Graphing Graph each system of equations by hand to find the solution. Then check by substituting into each equation.

1. 2.

Solution: Solution: Check: Check:

3. 4.


x

y

x

y

x

y

x

y



5. 6.


7. 8.


x

y

x

y

x

y

x

y



9. Which systems in 1–8 have no solution? Which have an infinite number of solutions? How

can you tell?



Systems with Equations in Different Forms For each system of equations: (1) graph the equations by hand to find the point of intersection and (2) rewrite the equations in slope-intercept form and use a graphing calculator to find the point of intersection.

1. 2.

Equations in Slope-Intercept Form: Equations in Slope-Intercept Form:

Solution: Solution: 3. 4.

Equations in Slope-Intercept Form: Equations in Slope-Intercept Form: Solution: Solution:

x

y

x

y

x

y

x

y

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SOLVING SYSTEMS OF LINEAR EQUATIONS BY SUBSTITUTION

CCSS: A-REI #5, A-REI #6 Overview In this investigation, students will use the substitution method to solve systems of linear equations. Students will learn that the underlying mathematical justification for the substitution technique is the substitution property of equality – that one can substitute equivalent expressions for each other. To date, they have experienced substituting a single value for a variable when evaluating algebraic expressions. In this investigation, students substitute algebraic expressions for a variable. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? • Students will solve a system of linear equations using the substitution

method. • Students will explain what the solution to a system of linear equations

means in the context of a real world problem. Assessment Strategies: How Will They Show What They Know?

• Exit Slip 6.2.1 requires students to solve a system of equations by substitution.

• Exit Slip 6.2.2 requires students to use a system of equations to find the break-even point.

• Journal Entry 1 asks students to explain the meaning of the word “substitution.”

• Journal Entry 2 asks students to explain how to use a table, a graph, or the substitution method to solve a system of equations.

Launch Notes Heifer International is a charitable organization that sends farm animals and seeds, and provides agricultural and business instruction, to people who do not have enough to eat. Two major tenets of the Heifer Project are “Pass on the Gift” whereby the receiver becomes a giver when new animals are born, and “Give a man a fish, he eats for a day. Teach a man to fish, he eats for a lifetime.”

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You may write to Heifer International or go to www.heifereducation.org in advance in order to have a classroom set of informational brochures, stickers and other materials on hand for the first day of the lesson. Start the lesson by inviting students to read about the Heifer International projects. Students can summarize and share information from the brochures and discuss the philosophy of the Heifer Project, what they do, how they do it, who donates, and who receives Heifer gifts. You may choose among other options for launching this investigation depending upon students’ interests and what they are studying in other courses. Some suggestions include:

1. Have students write an essay or discuss in groups either “Pass on the Gift” (which is also Pay it Forward from the book and movie of that name) or the saying “Give a man a fish….”

2. Show a video clip from Heifer International at www.Heifer.org or www.HeiferEducation.org.

3. Ask students to research Heifer International before class and write down two facts about the Heifer program to share with the class.

Closure Notes By the end of the lesson, students should know two techniques for solving a system of equations—graphing and substitution, and be able to describe the advantages of each. You may check their understanding with the two exit slips described below. Teaching Strategies

I. Begin the lesson by having the students learn about Heifer International, described above, and connect the scenario, as appropriate, to world hunger and nutrition, community service projects, world cultures, geography, or agriculture. Arrange students into pairs or small groups and have them answer Question 1 on Activity 6.2.1 Passing on the Gift. Do not tell the students how to work the problem. Allow them time to use the mathematical ideas they already know to solve the problem. The goal is to have students discover the substitution method for solving systems of equations. Students may use ideas from the previous investigation to solve the problem such as making numerical tables or writing algebraic equations. Ask probing questions and encourage students to develop a variety of methods for solving the problem. Some students may graph the information, some may write equations, and others may make tables of the following type:

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A table is a useful step in the transition from verbal statements to algebraic equations, so encourage its use, and, later, ask students if they can write the system of two equations. Since students will have just solved systems by graphing, some will use that method. Some students will be able to write two equations, and may figure out the substitution method on their own. You may walk around to each group and encourage different groups to solve the problem using different methods according to the group’s inclinations. If a group finishes early, you may ask that group to solve the problem in a different way – numerically, graphically, or algebraically depending on which method is not yet developed by the group. For example, if a group has solved the problem by graphing two lines, the teacher can ask the students to find algebraic equations for the lines and use the equations to find the solution. One group may show signs of using the substitution method for solving a system of equations. Ask probing questions to confirm students’ thinking. “What are the variables and how are they defined? What are the two equations you can write with the two variables? Could you solve one equation if it used only one variable? Can you change one equation so that one variable is written in terms of another variable?” Be sure to have students present to each other a variety of ways to attack the problem. If a group is close to solving the system using substitution, you may ask leading questions to help them discover the substitution method. During the whole class discussion, be sure the students articulate the idea that the point of intersection of the two lines is the point that makes both equations true. Elicit ideas about why graphing and numerical solutions might be cumbersome. Have a whole class discussion on the substitution method. Begin by asking them for examples of substitution in their experience. They may think of substitutes on the soccer field, substitute teachers, and cooking substitutions. A substitute is one thing that stands in for another. In math class students have substituted values for variables when evaluating expressions for given values. The substitution method for solving a system of equations involves finding an equivalent expression to

Number of Goats, g 10 9 8 7

Number of flocks of chickens, c 0 1 2 3

Total Cost $120(10)+

$20(0) = $1200

$120(9)+$20(1) =$1100

$120(8)+$20(2) =$1000

$120(7)+$20(3) =$900

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replace one variable in order to transform an equation in two variables to an equation in one variable.

Ask students to explain what the solution means in the context of the problem. Confirm that the various methods for solving the problem lead to the same result. Explain that the problem that they just solved is called a system of linear equations. Have students help define what a system of equations represents. If one group was able to solve by substitution, be sure to reference the work of that group, and say that we will now formalize that method. Otherwise, ask the class probing questions to lead them to the method. Discuss the essential requirement for substituting in mathematics: the substitute expression must be equivalent to the original expression. Also ask, “When have we seen substitution in math class?” Point the discussion in the direction of evaluating expressions with defined replacement variables (evaluate the expression 3x+2y for x=1 and y=–2) or the f(x) notation (Given f(x), evaluate f(2)). Develop the idea that expressions, not just numbers, can be substituted in for variables. On the board, you can write a system that uses geometric shapes or a nonsense word instead of the variables x and y. Show that we could substitute these shapes or nonsense words into the equation – though we have not defined operations on shapes or words leaving the operations undefined, we could write the expression. If you used a geometric shape, you could fill in the shape with a mathematical expression to show how an expression such as 5x + 7 can be substituted in for y. Thus, you can help the students become visually comfortable with substituting multiple part expressions in for a single variable. Emphasize the mathematical concept underlying this method -- “the substitution property of equality” that states “equivalent expressions can be substituted for each other”, or “If a = b, then “a” can be substituted for “b” in a mathematical expression.” Ask students to identify different manifestations of the substitution property of equality that have already occurred in their math experience.

In a whole class guided discussion, work through the steps involved in solving a system of equations by substitution. Have students write process cards to place in their math toolkit or the “Formula Reference” section of their notebooks.

Next students may work in pairs to tackle the other two problems in Activity 6.2.1 Passing on the Gift using substitution. As you circulate, note which pairs solve using one variable or another or one equation or another. Be sure that the pairs which show different approaches present their solution to the class, so that the class can see that it does not matter which variable you choose to solve for, nor which equation you use to solve for the variable. It is likely that some students will try to substitute the variable they solve for back into the same equation. Explain that this leads to an identity, does not utilize the other equation in the system, and does not lead to a solution of the system. If any of these variations do not present themselves, direct one of the faster pairs to try out what would happen if, for example, you solved for a variable in the first equation, and then substituted

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the found expression back into the first equation. Ask another pair to try solving for the other variable first.

Some students may use only one variable to identify the two pieces of information being asked for. For example, in the first problem, “Let g = the number of goats, and let 10 – g be the number of flocks of chickens”. Note that this method of solving problems would give the equation 120g + 20(10–g) = 1000 and is mathematically equivalent to the substitution method for solving a system of equations if we let f = number of flocks, solve for f in f + g = 10 and substitute for f in the equation 120g + 20f = 1000. Be sure the students see that the mathematics is the same regardless of the approach. Comparing the two approaches shows that by judicious naming of variables, one can bypass the step where one solves for a variable in one equation to substitute into the other equation. Using two variables, however, has the advantage of being more transparent in setting up the equations. Insist that the students learn to write a system of equations in this unit. Wrap up by reviewing the steps in solving a system using the substitution method, noting that it is easier to solve for a variable when the coefficient is one. Discuss the pros and cons of solving a system algebraically compared with graphically.

You may use Exit Slip 6.2.1 requiring the students to solve a system of equations by substitution. Activity 6.2.2 Solving Systems by the Substitution Method may be used as homework. Additional practice is provided in Activity 6.2.3 More Practice with the Substitution Method, which may be used as needed.

Differentiated Instruction (For Learners Needing More Help) Students may need some guidance in identifying the two variables and writing the two equations. Underline the words that describe the variables and translate the given information into equations. Have the students write the solution as an ordered pair, and write in words what the solution means in the context of the problem. You may allow students to use the “Formula Reference Section” from their notebooks, which could include a procedure card on how to solve a system by substitution.

Group Work Have students work in pairs to solve problems 2 and 3 on Activity 6.2.1. Hopefully student pairs will raise questions such as whether it matters which equation is used to solve for a variable or which variable is solved for. Have them try the problem one way then the other way and compare results.

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Journal Entry 1 Look up the word “substitution” in the dictionary and explain why this method is called the “substitution method.”

II. Activity 6.2.4 Drag Racing yields a system of equations where both equations are in slope-intercept form. Students will receive practice using variables other than x and y. Note that the car starts at the 0-meter mark on the track and the truck starts at the 120- meter mark. Help students to see that the slope is the rate of change or speed of the vehicle in meters per second and the starting point on the track is the y-intercept. The equations will be d = 30t + 120 and d = 45t. In solving a system of equations with both equations in the form y = mx + b, one approach is to substitute the second expression for y in for the y in the first equation, thereby “setting the y’s equal to each other” and then solving the one resulting equation in x. In fact, in any system, students may prefer to solve both equations for y, set the equations equal to each other and solve. An advantage to this approach is the ease in checking the answer with the graphing method on a calculator. You can set the stage for The Drag Racing Activity by asking students if they have ever gone to a car race. Drag racing videos are available at http://www.dragtimes.com/drag-racing-videos.php or by searching “drag race videos”. Emphasize that drag racing on public streets is illegal and deadly for not only the racers, but for innocent bystanders, too. “Would it be fair for a pickup truck to race a Mustang? How can you make the race fair if one vehicle is much faster than the other?” “To be Fair” could be defined as “each person has an equal opportunity to win”. Discuss giving a head start to the slower vehicle. In this activity the head start will take the form of the slower vehicle starting 120 meters closer to the finish line. (This is mathematically equivalent to giving the truck a 4-second head start).

There is an applet simulating the drag race that you can show to the class. Go to http://uhaweb.hartford.edu/rdecker/ and click on the link to Algebra Curriculum Project. Directions for using the applet are under the Applet Help menu. Click on and move the slider in the box below the graphs to see the race on the left and the corresponding time-distance plot on the right. Point out to the class how the applet helps them visualize various events along the race such as “Where does the Mustang overtake the pickup?” and “How far ahead is the Mustang at the end of the race?” Students may finish the worksheet for homework.

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Differentiated Instruction (For Learners Needing More Help) Students could walk a simulation of a race. If you haven’t already done so in earlier units, use a motion detector to show that the time distance graph of a person walking at a constant rate is a straight line. Contrast this with a person speeding up (concave up graph) or slowing down (concave down graph). As an extension, students might use the race applet to change the parameters of the problem by using different starting points or different speeds. Ask students to predict the results and then check using the applet or the graphing calculator.

Group Work Have students begin work on Activity 6.2.4 in groups. Review student work as you circulate among the groups. Be sure everyone has the equations d = 45t and d = 30t + 120 before they leave the class. Note that solving a system of equations by setting the y’s equal to each other is an example of the transitive property of equality: if 45x = y and y = 35x + 120, then 45x = 35x + 120. (If a = b and b = c, then a = c.)

Journal Entry 2 If you have a system of equations, explain how you would solve the system using: a) a table b) a graph c) substitution method

III. In Activity 6.2.5 Break-Even Analysis, students create a system of equations in

which both equations are in slope-intercept form. In this example, however, the dependent variables are not the same: one is total cost and the other is revenue. The break-even point is where revenue equals cost so we set the y’s (i.e. the dependent variables) equal to each other.” To launch this activity, you can ask students if they have ever participated in a fundraiser or business venture. Have the students identify the fixed costs, variable costs and revenue for the situations the students describe. Students should learn the vocabulary of revenue, cost, variable cost, fixed cost and break-even point. Have the students summarize the information by writing semi-algebraic expressions such as:

!"#"$%" = !"#$% !"# !"#$ !"#$%& !" !"#$% !"#$ !"#$ = !"#$"%&' !"#$ + !"#$% !"#$

The break-even point occurs is when !"#"$%" = !"#$.

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Call the class to attention after about 10 minutes to be sure everyone has the correct total cost function C(x) = 0.15x + 450 and revenue function R(x) = 1x. Highlight the different ways students arrived at the equations. Did they make tables? Did they just fill in for slope and y-intercept in the slope intercept form of an equation? Why is the intercept for the revenue equation zero? Once the students understand how to develop the cost and revenue equations, have them continue working on the Popcorn problem in their groups. When everyone has completed Activity 6.2.5, have one group present their solution. Discuss the advantages or disadvantages of the algebraic substitution method over the graphical method. In the solution, students should realize that the break-even point is when Revenue = Cost, and write R(x) = C(x) on the board. Then replace R(x) with an equivalent expression 1x. Similarly, replace C(x) with 0.15x + 450. You now have 1x = 0.15x + 450. The solution, x ≈ 529.41 is found with greater precision using the substitution method than with a graph. In the real world context, however, students should realize that a fractional bag of popcorn does not make sense. They will have to sell 530 bags in order to break even. Note that Question 10 requires representing an inequality on a number line as introduced in Unit 2.

Differentiated Instruction (Enrichment) Some students may want to talk about profit (see Activity 6.2.5). Observe that Profit = Revenue –Cost or P = R – C. Since the break-even point is where Profit is 0, substitute 0 in for P: 0 = R – C which becomes: R = C. If profit is 0 when R = C, ask when profits will be positive (when R>C) and when one operates at a loss (R<C). One could find the break-even point by finding a profit function P = 1x – (0.15x + 450) and solving for P = 0. Graph all three equations on the graphing calculator: profit, revenue and cost. Note that the x-intercept of profit function is the same x value as the intersection of the revenue and cost functions. Notice that the interval on which the cost function is greater (higher) than the revenue function is where the profit is negative (the business is losing money) and the interval where revenue is greater than cost is where the profit function is positive.

Group Work

Activity 6.2.5 is appropriate for heterogeneous groups. Although there is a lot of new vocabulary (cost, revenue, break-even point, etc.), the more able students should be able to help their partners figure out what to do. Students can finish the break-even analysis problem on Activity Sheet 6.2.5 for homework. You may also create a few homework problems that use topics of student interest, and/or have the students create systems of equations to solve an interesting problem. This is a good opportunity to personalize the problems for the students.

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Exit Slip 6.2.2 presents another situation in which students must find the break-even point.

IV. Activity 6.2.6 Systems of Equations in Slope-Intercept Form includes several systems of equations that have equations in slope-intercept form. The systems involve negative integers. Two systems have a single solution, one has no solution, and one has infinitely many solutions. This provides a review of the concepts from the first investigation. You might have the class come together as a whole. Ask the students as a class to describe what happens algebraically when you try to solve a system with no solution (a patently false statement results) and when you try to solve a system with infinitely many solutions (a patently true statement results). Ask the students to describe how they can tell whether there is no solution before they even begin to graph or solve the system algebraically. (The slopes of the two lines are equal, but the y-intercepts are different.) Have the students check their work by graphing the system on a graphing calculator. Give students a few minutes to write down any questions or concerns they have about solving a system of equations by the substitution method. Ask students to share what they wrote.

Differentiated Instruction (For Learners Needing More Help) If the students need more practice transforming verbal descriptions to algebraic equations, you might have them work on three different scenarios that result in identical equations. Point out that the math is independent of the real-world contexts, which is part of the power of mathematics as a problem-solving tool. Activity 6.2.7 One (Equation) for All and All for One is an example of a set of problems with different contexts, but identical equations.


• Activity 6.2.1 Passing on the Gift • Activity 6.2.2 Solving Systems Using Substitution • Activity 6.2.3 More Practice with the Substitution Method • Activity 6.2.4 Drag Racing • Activity 6.2.5 Break-Even Analysis • Activity 6.2.6 Solutions to Systems of Equations • Activity 6.2.7 One for All • Exit Slip 6.2.1 Substitution Method • Exit Slip 6.2.2 Breaking Even • Bulletin board for key concepts • Graphing Calculators • Student Journals



Passing on the Gift One of the major tenets of the Heifer Project is “Passing on the Gift”. In this lesson you will learn what gifts cost, and how many gifts will be paid forward next year. 1. Village Gardens in Portland, Oregon is a group of public housing residents, many of whom

are refugees or immigrants. Heifer International will provide the group goats for milk and chickens for eggs. Each goat costs $120 and each flock of chickens cost $20. Portland’s middle school and high school musicians raised $1,000 by putting on a charity band and orchestra concert for Village Gardens. The village has requested 10 gifts, so the total number of goats and flocks of chicken will be 10. How many goats and flocks of chickens will be sent to the village? a. Define the variables in this problem.

b. Develop an equation that represents the number of goats and chickens.

c. Develop an equation that represents the cost of the goats and chickens.

d. Solve the system of equations using the substitution method.

2. Eight animals will be sent to Cameroon. The animals will be cows and pigs. A cow costs

$500 and a pig costs $120. The total cost of all animals is $2,100. Find how many of each animal will be sent to Cameroon. a. Define the variables in this problem.

b. Develop an equation that represents the number of cows and pigs.

c. Develop an equation that represents the cost of the cows and pigs.



(Problem 2 continued)


3. This year, a family in Kosovo that received goats and chickens from Heifer plans to breed their animals to produce 351 new animals to share with their neighbors. One goat produces 3 offspring per year, and one chicken produces 42 chicks a year. In addition to breeding the animals, the family will gain 88 servings of milk and eggs from the animals each day. One goat will yield 16 cups of milk per day, and one chicken will produce 1 egg a day. A serving consists of one cup of milk or one egg. The milk that they don’t drink may be made into yogurt and cheese. Any eggs or milk that they don’t use may be given to a food bank or sold to supplement family income. How many goats does the family receive from Heifer to start the breeding program? How many chickens did the family have at the start? a. Define the variables in this problem.

b. Develop an equation that represents the number of offspring from the goats and chickens each year.

c. Develop an equation that represents the number of servings of food from the goats and chickens.




Solving Systems by the Substitution Method 1. Solve the following systems of equations by the substitution method. Show your work.

a. 8x + 5y = !14

y = !3x

b. 6x ! 4y = 38 x + y = 5

c. x ! 3y = 9 6x ! 5y = 2



2. To eliminate the problem of World Hunger we need to understand what constitutes nutritious meals. This is important in order to ensure that people eat the food they need to stay healthy. The average 128-pound person needs to consume about 36 grams of protein and 2,200 calories each day.

An egg has 6 grams of protein and 77 calories. A 6-ounce glass of goat’s milk has 8 grams of protein and 168 calories. If a person has enough eggs and goat’s milk each day, they will meet the required 36 grams of protein per day. How many eggs and how many 6-ounce glasses of goat’s milk will a person need to eat each day if they want their total protein intake to be 36 grams and their total calorie intake from these two sources to be 658 calories? (Assume that there are other foods available that do not contain protein but will supply the remaining calories needed.)

a. Develop an equation that represents the number of calories consumed from eggs and goat’s milk each day.

b. Develop an equation that represents the amount of protein consumer from eggs and goat’s milk each day.

c. Solve the system of equations using the substitution method. Explain what your solution represents in the context of the problem.



3. A serving of rice contains 251 calories, and a serving of beans has 227 calories. How many servings of rice and how many servings of beans will provide a total of 2,187 calories if the person eats twice as many servings of rice per day as beans? a. Define your variables and then write an equation that represents the number of calories

consumed from rice and beans each day.

b. Write an equation that represents the number of servings of rice and beans consumed each day.

c. Solve the system of equations using the substitution method. Explain what your solution represents in the context of the problem.



More Practice with the Substitution Method Solve each system of equations using the substitution method.

1. 3 15

3+ =

=x yy

2. 4 20

4− =

=a ba

3. 3 23

10c d

d+ =

= − 4.

23 5

=+ =j kj k

5. 6 5

5− = −

= +m n

n m 6.

3 2 129

− =− + =g hg h



7. The perimeter of Mrs. McCord's rectangular garden is 100 feet. This can be represented with the equation 2 2 100+ =w l , where w is the width of the garden (in feet) and l is the length of the garden (in feet). If the length of the garden is 1.5 times longer than the width ( 1.5=l w ), what are the dimensions of the garden?

8. Is (5,6) a solution to the following system? How do you know?

3 2 27

1+ =

= +x y

x y



Drag Racing

The race car drivers of a 2004 Dodge Ram pickup truck and a 1965 Ford Mustang are scheduled to race at the NY International Raceway Park. On a typical straight-line race track, the average speed of the Pickup is 30 meters per second and the Mustang is 45 meters per second. Because the Mustang is so much faster than the pickup, the drivers agree that the pickup will start 120 meters closer to the finish line, as shown below.

0 m 120m Finish 1. Create a table of values for the time in seconds (t) and distance

from the starting line in meters (d) for every two seconds. Assume that both are traveling at a constant rate, the vehicles do not speed up or slow down.

2. For the data in the table, write two equations in slope intercept

form (y = mx + b) that relate distance from the zero meter mark to the time spent running.

a. The equation for the Mustang’s distance as it relates to time is:

b. The equation for the pickup’s distance as it relates to time is:

c. Graph each of the equations on the same coordinate axes.

d. Label each line and axes appropriately.

e. Solve the system of equations from (1) and (2) using substitution.

Time (t)

in sec

Distance (d) in meters from the start point

Mustang Pickup 0 2 4 6 8 10 12



f. How long must the track be for the pickup to win the race?

Answer the following questions using complete sentences and show your work. 3. If the track is 600 meters long, which vehicle will win the race?

Describe how you can determine the answer to question 3:

a. from a table b. from a graph c. from a system of equations

4. If the Mustang gives the pickup a 210-meter head-start, which vehicle would win the race? (Assume

the same track as in Question 3.)

Show how you can answer this question three different ways:

a. Use a table. b. Draw a graph.

Time (sec)

Distance (m)

Mustang Pickup



c. Solve a system of equations.

5. If the Mustang and Pickup could race forever at the same speed, will the Mustang eventually overtake

the pickup, no matter how much of a head-start the pickup receives? What part of the linear equations tells you? (Assume that you are not confined to a 600 meter race.)



Break-Even Analysis: Popcorn The business club is going to sell popcorn at hockey games. Since they are astute business men and business women, they know that they will not make a profit right away because they have to pay the cost of buying a popcorn machine. They need to know how many bags of popcorn they must sell in order to cover the set-up costs. In other words, what is the break-even point for their popcorn business? The red and glass popcorn carts often seen at carnivals and fairs costs $450. This is the fixed cost. Regardless of the number of bags of popcorn they make and sell, the machine cost will not change.

1. The popcorn, butter, salt, and serving bags cost $15 for every 100 bags of popcorn. What is

the cost per bag for these consumables? The variable cost changes depending on how many bags of popcorn they make. The more popcorn they make, the more they spend on popcorn, butter, salt and bags. The variable cost is $0.15 times the number of bags of popcorn. The total cost is the variable cost plus the fixed cost.

2. Write an equation for the Total Cost as a function of the number of bags of popcorn made.

Use the notation C(x) for total cost, and let x be the number of bags of popcorn they make.

Each bag of popcorn sells for $1.00. The revenue is the amount of money they receive from selling bags of popcorn. If they sell 20 bags of popcorn, they will receive $20, since each bag sells for $1. The revenue they take in is the price per bag of popcorn multiplied by the number of bags of popcorn sold.

3. Write an equation for the Revenue as a function of the number of bags of popcorn sold. Label

the revenue function R(x).



The break-even point occurs when the amount of money they receive from selling popcorn is equal to the amount of money they spent to make the popcorn. It is when Revenue = Total Cost. The break-even point tells how many items they must create and sell in order to recover their expenses.

4. Take the Total Cost and Revenue functions that you developed above, and sketch the graph

of the two functions on one coordinate plane. Label the axes appropriately.

5. Estimate the break-even point graphically. 6. To find the break-even point algebraically, write R(x) = C(x).

7. Solve the equation R(x) = C(x) for x.

6000 100 200 300 400 500

600

0

100

200

300

400

500



8. Check your graphical estimate with your algebraic solution. Explain any difference.

9. Now that you found x, what does it mean in terms of the popcorn business?

10. The business will earn a profit when revenue is greater than total cost.

a. Use an inequality to represent the number of bags of popcorn that must be made and sold

to make a profit.

b. Show on a number line the number of bags of popcorn that must be made and sold to make a profit?



Solutions to Systems of Equations

1. Solve this system of equations using each method below.

2x + y = 5

2y = 10 – 4x

a. With a graphing calculator:

b. Using the substitution method:

c. Use your result from question 1a to explain why you obtained the result in question 1b.

2. Solve this system of linear equations using each method given.

y – 8 =2(x – 6)

y = 2x + 7

a. With a graphing calculator:

b. Using the substitution method:

c. Use your result from question 2a to explain why you obtained the result in question 2b.



3. Explain how you can tell from a graph whether a system of equations has one solution, many

solutions, or no solution. 4. Which system of equations in questions 1 or 2 is inconsistent? Which system is dependent? Explain.

5. Suppose you obtain the following results in solving three different systems of linear equations using the substitution method. How many solutions does each system have?

a. 5 = 5 b. –6 = 4 c. y = 8

6. Create a system of linear equations that has no solution. 7. Create a system of linear equations that has an infinite number of solutions.



One (Equation) for All and All for One

1. A magician has a magic trick that uses an 18” length of string that is cut into two pieces. One

piece is 2 inches longer than the other. Find the length of each piece.

2. The Interscholastic Math Team has 18 members, all freshmen and sophomores. There are two more freshman than sophomores. Find the number of freshman and the number of sophomores.

3. Find two numbers such that one number is 2 more than the other and their sum is 18.

4. What do you notice that is similar about the three problems above?

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SOLVING SYSTEMS OF LINEAR EQUATIONS BY ELIMINATION

CCSS: A-REI #5 Overview In this investigation, students will use the elimination method to solve systems of linear equations, identify the characteristics of a system of linear equations that lend themselves to the elimination method, and interpret the solution of a system of linear equations within the context of the problem. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? • Students will use the elimination method to solve a system of equations. • Students will explain the algebraic properties upon which the elimination method

is based. • Students will explain the relationship between the number of solutions to a system

of equations and the relationship between the slopes and y-intercepts of the equations within a system.

• Students will identify the characteristics of systems of equations that lend themselves to the substitution and elimination methods.

Assessment Strategies: How Will They Show What They Know?

• Exit Slip 6.3.1 requires students to solve a system by elimination and identify the algebraic properties used.

• Journal Entry 1 asks students to explain when the multiplication property must be used to solve a system by the elimination method.

• Journal Entry 2 asks students to explain how graphical features correspond to algebraic solutions of simultaneous equations.

• Exit Slip 6.3.2 requires students to compare different methods of solving systems and to explain their choice of method in solving a particular system.

Launch Notes As a launch to this investigation you might show the 5 minute Computer Problems vignette. Go to http://www.thefutureschannel.com/hands-on_math/computer_problems.php from The Futures Channel which introduces students to the Dell Computers call center in Round Rock, Texas. The video shows the different facets involved in ordering and assembling a computer from component picking to shipping. Based on information in the video, you and the class may develop a number of simultaneous equation scenarios based around cost and time constraints to use as part of this

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investigation. For example, the shipping center produces 100,000 computers each day. Suppose some of them sell for $500 and the remainder sell for $800. If the total income for the day is $62,000,000 how many of each type were sold? You may pose this question and then answer it at the end of the first day of the investigation. Closure Notes The investigation culminates with students successfully demonstrating their ability to solve a system of equations, identify the method that they used, and explain why they chose that particular method. Students will identify the main characteristics of systems of equations that lend themselves to the substitution and elimination methods, respectively. This may be done with Exit Slip 6.3.2, followed by a class discussion. Teaching Strategies

I. The focus of this investigation is the elimination method for solving systems of linear equations in two variables. Students should understand how this method utilizes the addition and multiplication properties of equality. Students should learn to recognize which method is most appropriate when solving a system. Begin with an example that leads to a simple system such as: The sum of two numbers is 20. The difference between them is 2. Find the numbers. The following system can be written:

x + y = 20 x – y = 2

where x is the larger of the two numbers and y is the smaller of the two numbers. Have students add the equations using the addition property of equality to see that the y variable is eliminated. Give further examples where the x or y variable is eliminated through simple addition, such as:

x + 2y = 5 and x – 2y = 7 – x + y = 7 3x + 2y = 13. Once students see the benefit of being able to eliminate one variable in a system of equations, supply examples where multiplication must be done to at least one equation in the system before the addition of the system results in the elimination of a variable. An example of this is the following system:

5x – 2y = 19 3x + y = 7.

To help students develop an understanding of the sequence of steps involved in the elimination method, allow them to add a system like this before any changes are made and see that neither variable is eliminated. Ask students to explain why neither variable was eliminated and what is different about this system than the previous ones that cause this to

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happen. Emphasize during the discussion the idea that when solving an equation what is done to one side of an equation must be done to the other side for the equation to remain in balance and the solution of the equation to remain unchanged. Therefore, it is not possible to simply “insert” a coefficient of 2 in the second equation of this system. Students should recognize that the elimination method can be justified in terms of the multiplication property of equality in conjunction with the addition property of equality. Emphasize, as you did with the substitution method, that the values of both variables satisfy both equations, and thus provide the solution to the system.

You may use Activity 6.3.1 Introduction to the Elimination Method in class or for homework to help students learn the elimination method. Questions 5–7 are designed to help students see the role of the least common multiple when using the elimination method in cases where both equations need to be transformed and to make the connection to the addition of fractions with unlike denominators. You may now return to the problem posed in the launch. Let x represent the number of $500 computers sold and y the number of $800 computers sold. The system of equations is:

x + y = 100,000 500x + 800y = 62,000,000.

We can eliminate x by multiplying the first equation by –500 and adding the result to the second equation. We then find that y = 40,000 and x = 60,000. You may use Exit Slip 6.3.1 which asks students to solve a system of equations using the elimination method and explain the properties that are used in the solution.

Journal Entry 1 When solving a system of equations with the elimination method, how do you know whether it is necessary to multiply both sides of an equation by a number before adding it to the other equation?

Differentiated Instruction (For Learners Needing More Help)

Exploration and discussion of the elimination method should focus on both the process used as well as the algebraic principles that support the process. If students understand the algebra that supports the elimination method but have difficulty remembering and/or following the sequence of steps involved, you might have them work in pairs or small groups to develop a note card that describes the sequence of steps in their own words and includes one or more examples of how to implement the elimination method. Use simple equation pairs, such as x + y = 12 and x – y = 16, as examples.

II. Provide explorations that use the elimination method to solve systems of equations that have no solution, one solution, or many solutions (for example: 2x + 8y = 6 and x + 4y = 3). To emphasize the connection between the number of solutions to a system of equations

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and the number of intersections between the lines, write the two equations in the slope-intercept form and then graph the equations. By putting both equations in slope-intercept form, students can readily identify the slope and y-intercept of the lines, graph the lines as needed, and make connections between the graphical and algebraic nature of the lines. Activity 6.3.2 Exploring the Number of Solutions may be used to help students discover the connection between the number of solutions and the nature of the graphs.

At this point you may want to assign students some or all of the applications problems in Activity 6.3.3 Applications of the Elimination Method. Note that question 6 is open-ended and depends upon what students consider to be a “reasonable profit.”

Differentiated Instruction (Enrichment) Assign students question 6 in Activity 6.3.3 and then ask them to create one or more additional open ended problem that can be solved using a system of equations.

Have students complete Exit Slip 6.3.2, which requires that they explain graphical characteristics of systems of equations that have none, one, or an infinite number of solutions.

Journal Entry 2 Explain what happens when you solve a system of equations when the graphs of the equations are two parallel lines, when the graphs of the equations are two intersecting lines, and when the graphs of the equations are the same line.

III. Once students understand the algebraic principles that support the elimination method and

can clearly describe the relationship between the number of solutions and the number of intersections of the graphs, you may lead students in an exploration of the characteristics of systems of equations that lend themselves to the substitution method versus the elimination method. You may use systems of equations students have already solved in Investigations 1 and 2, or additional examples from Activity 6.3.3 Applications of the Elimination Method. Ask students to work independently, in pairs, or in small groups to identify characteristics of systems of equations that lend themselves to each method. When identifying common characteristics it is important to remind students that they may use either the elimination or substitution method. However, by looking at the characteristics of the equations within a system, one method might be easier to work with then the other. Students may design a graphic organizer to list the characteristics. For practice using the elimination method, and for getting students to think about the steps that are required, you may use Activity 6.3.4 Mechanics of the Elimination Method. Activity 6.3.5 Selecting an Algebraic Method gives students practice choosing a method and explaining their choice.

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Group Activity Have students work in groups of three. Give each group a system of equations. Have one student solve the system by graphing, one by substitution, and one by elimination. Have them check each other’s work.


• Activity 6.3.1 Introduction to the Elimination Method • Activity 6.3.2 Exploring the Number of Solutions • Activity 6.3.3 Applications of the Elimination Method • Activity 6.3.4 Mechanics of the Elimination Method • Activity 6.3.5 Selecting an Algebraic Method • Exit Slip 6.3.1 Elimination Method • Exit Slip 6.3.2 Solving Systems • Bulletin board for key concepts • Student journals



20-5 5 10 15

20

-5

5

10

15

Introduction to the Elimination Method 1. Solve the following problem: The sum of two numbers is 20, and their difference is 2. Find

the two numbers. a. The system of equations below models

this problem if ____ represents the larger

number and ___ represents the smaller

number.

x + y = 20 x – y = 2

b. In the space above, add the two equations together (left side added to left side, right side added to right side). You now have an equation in one variable. Which variable is it, x or y?

c. Solve for the variable in part (b). Then substitute it into one of the original equations to solve for the other variable.

d. Graph both equations by using the x-intercept and y-intercept on the coordinate plane above. Identify the intersection point.

2. Solve each of these systems by adding the two equations together to eliminate one of the

variables.

a. –x + 2y = 5 b. x – 2y = 7 x + y = 7 3x + 2y = 13



3. Consider the following system of equations: 5x – 2y = 19 3x + y = 7

a. What happens when you add these equations together? Is one of the variables eliminated? Show your work below.

b. Go back to the initial system. Multiply both sides of the second equation by 2. Now add this new equation to the first equation. What happens?

c. Complete the solution to this system of equations. 4. Write a system of equations for this problem and solve it by elimination:

The sum of two numbers is 12. When 3 times the smaller number is subtracted from the larger number, the result is 4. Find the two numbers.



5. Find the sum of these two fractions: !!+ !

!.

6. Solve this system by first eliminating the variable y. (Hint: You will have to find one

number to multiply the first equation by and a second number to multiply the second equation by.) 2x + 3y = 9 5x – 4y = 11

7. How are questions 5 and 6 related?

8. Explain how you used the addition and multiplication properties of equality to solve systems of equations using the elimination method.



Exploring the Number of Solutions

Solve each system of equations using the elimination method. (Be prepared for some surprises!) Then graph the system. 1. 3x + 2y = 24 x + 2y = 20

2. x + y = 4 3x + 3y = 18 3. –5x –3y = 30 10x + 6y = – 60

-5 5

-5

5

-5 5

-5

5

-5 5

-5

5



4. Describe what the graphs look like if your system has

a. No solution

b. One solution

c. An infinite number of solutions

5. Describe what happens when you eliminate one variable if the system has:

a. No solution

b. One solution

c. An infinite number of solutions

6. Which system in Questions 1–3 is inconsistent? Which system is dependent? Explain.



Applications of the Elimination Method For each problem, write a system of linear equations and solve them using the elimination method. 1. You are planning a picnic for Memorial Day. You need to buy enough hot dogs and

hamburgers so that each of your 10 guests can have two servings. You determined that hot dogs cost $0.40 each and hamburgers cost $0.80 each. You have $12 to spend on the hamburgers and hot dogs. How many hot dogs and how many hamburgers should you buy?

2. Your family is planning to take the Amtrak train from Hartford to New York City for a day

trip. As a result of some research you learn that your friend Jackie took the train with a group of 3 adults and 5 children and it cost them $269.50. A cousin also took the train to the city with a group of 2 adults and 3 children and it cost them $171.50. Find the price of an adult’s ticket and the price of a child’s ticket.



3. During the 2008-2009 basketball season the UConn women’s team had an incredible undefeated season (39-0) and won the NCAA championship. Maya Moore and Renee Montgomery were the top scorers during the year and together they scored 1,398 points. If Maya scored 110 more points than Renee, how many points did each player score during the season?

4. During the 2008-2009 men’s basketball season, UConn’s Hasheem Thabeet and Jeff Adrien

had a total of 746 rebounds. Jeff had 30 fewer rebounds than Hasheem. How many rebounds did Hasheem and Jeff each have during the season?



5. At the upcoming school fair, your class is planning to raise money for a class trip to Washington, DC. You plan to sell your own version of Connecticut Trail Mix. After doing research on the cost of various ingredients, you find you can purchase a mixture of dried fruit for $3.25 per pound and a nut mixture for $5.50 per pound. The class plans to combine the dried fruit and nuts to make their unique Connecticut Trail Mix that sells for $4.00 per pound. After researching the number of people who attended last year’s fair, you anticipate you will need 110 pounds of trail mix. Suppose the cost of making 110 pounds is exactly equal to the revenue from selling the trail mix. How many pound of dried fruit and how many pounds of mix nuts were used?

6. In question 5, how many pounds will you need to sell in order to make a reasonable profit?

Explain your reasoning. 7. Find a system of equations that you solved in Investigation 2 and now solve it using the

elimination method.



Mechanics of the Elimination Method

(Problems 1–3) The elimination method for solving a system of two linear equations works if the two equations are both in standard form so that the x-terms, the y-terms, and the constant terms line up with each other. In each of these systems, rewrite the second equation so that it “lines up” with the first one.

1. 3x + 2y = 10 y = 2x – 9

2. 4x + 2y = 6 3y – 2x = 25

3. x + 10y = 0 –15 = 5y + 2x

(Problems 4–6) Once the equations are lined up, you can sometimes eliminate a variable by adding the two equations together. For which system can you eliminate x by adding? For which system can you eliminate y by adding? For which system will neither variable be eliminated when the equations are added?

4. 2x – y = –2 3x + y = 17

5. x + 2y = 9 –2x + y = –8

6. 4x – 2y = 16 – 4x + 8y = 8

(Problems 7–9) Sometimes you can eliminate a variable by multiplying one of the equations by the same number on both sides and then adding it to the other. For each of these systems, which equation should be multiplied and by what number? Which variable will be eliminated when you do that?

7. 2x + y = 2 3x – 2y = –11

8. –x + 2y = –7 3x + 5y = –1

9. x + 3y = 5 2x + 4y = 7



(Problems 10–15) Complete the solutions to the systems in Problems 4–9.

10(4). 2x – y = –2 3x + y = 17

11(5). x + 2y = 9 –2x + y = –8

12(6). 4x – 2y = 16 – 4x + 8y = 8

13(7). 2x + y = 2 3x – 2y = –11

14(8). –x + 2y = –7 3x + 5y = –1

15(9). x + 3y = 5 2x + 4y = 7



(Problems 16–18) Sometimes you need to multiply both equations in order to eliminate one variable. Figure out which variable you want to eliminate and then chose what to multiply each equation by. Explain your choices. 16. 2x + 5y = 45 3x – 2y = 20

17. 2x + 3y = 30 –3x – 4y = –41

18. 0.2x + 0.7y = 0.9 0.5x + 0.3y = 0.8

(Problems 19–21) Complete the solution to each of the systems in Problems 16–18.

19(16). 2x + 5y = 45 3x – 2y = 20

20(17). 2x + 3y = 30 –3x – 4y = –41

21(18). 0.2x + 0.7y = 0.9 0.5x + 0.3y = 0.8



Selecting an Algebraic Method

Solve each system of equations using either the substitution or elimination method. Explain your choice for each system. 1. y = 4x + 3 2. 2x – y = 11 y = – 3x – 2 8y – 2x = 52

Explanation: Explanation:

3. 2x – y = –10 4. 8x + 8y = 24 5x + 2y = 43 x = –5y + 11

Explanation: Explanation:

unit 6: systems of linear equations (3...

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