Linear Equations. 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber

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1 Linear Equations 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber Tools: Geometer s Sketchpad Software Overhead projector with TI- 83 adapter Math Book Graph Paper TI-83 plus Calculators Computers Calculator-Based Ranger (CBR 2) Green Globs Software Pencils Paper Journals

2 2 Unit Objectives Given a five day instruction on linear equations, the following are unit objectives: Given instruction on how to create a table of values on the calculator from a linear equation, students will demonstrate how to make a table of values. Given technology instruction on how to use Geometers Sketchpad, students will be able to graph linear equations in Geometers Sketchpad. Given a technical briefing on the TI-83 calculator, the students will be able to graph linear equations with their calculators. Students will calculate the x and y intercepts on the graph given linear equations. Students will create graphs and describe their slopes given instruction on the CBR. Students will be able to find the slope of a line. Students will be able to describe slopes with appropriate mathematical vocabulary. Given the TI-83 calculators, students will be able to find the slope. Given the slope-intercept formula, the students will be able to rewrite linear equations into the slope-intercept form. Using the Green Globs software the students will be able to write linear equations. Given the slope and a point on the line, students will be able to write linear equations. Given a technical briefing of the TI-83 calculator, students will be able to enter system of equations into the calculator. Given instruction on the matrix program of the TI-83 calculator, students will be able to solve systems of equations. Students will be able to explore equations on the calculator by graphing the linear equations. New York State Standards

3 3 Standard 3: Mathematics A.PS.4 Use multiple representations to represent and explain problem situations. A.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, function, equations, charts, graphs, Venn diagrams, and other diagrams. A.R.3 Use representation as a tool for exploring and understanding mathematical ideas. A.G.9 Solve system of linear equations graphically. A.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equation, or objects created using technology as representation of mathematical concepts. A.G.4 Identify and graph linear functions Given a technical briefing on the TI-83 calculator, the students will be able to graph linear equations. A.PS.4 Use multiple representations to represent and explain problem situations. A.A.33 Determine the slope of a line, given the coordinates of two points on the line. A.A.34 Write the equation of a line, given its slope and the coordinates of a point on the line. NCTM Standards Create and use representations to organize, record, and communicate mathematical ideas. Use Mathematical models to represent and understand quantitative relationships. Understand patterns, relations, and functions. Organize and consolidate their mathematical thinking through communication. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. Analyze and evaluate the mathematical thinking and strategies of others. Represent and analyze mathematical situations and structures using algebraic symbols. Apply and adapt a variety of appropriate strategies to solve problems. Resources

4 4 McDougal Littell, Math Course 3, by Boswell, Laurie, Kanold, Timothy, Larson, Ron, Stiff, Lee, Houghton Mifflin, Chapter 11, pages , copyright Getting Started with the CBR 2 Sonic Motion Detector, Texas Instruments, Holden Custom Products, pages 10-13, copyright Graphing Linear Equations, Elizabeth Stapel. Texas Instruments TI-84 Plus Silver Edition Guidebook, Texas Instruments, Banta Book Group, pages chapter 1-2, 1-93, copyright 2004 Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, chapter 7, pages , copyright Mathematics Core Curriculum MST Standard 3, State Education Department, pages , copyright 2005 Materials Used Daily

5 5 Day 1: Geometer s Sketchpad Class work worksheet Paper Pencil Journals Day 2: TI-83 plus Calculator Overhead with TI adapter Instructional worksheet Paper Pencil Journals Day 3: Calculator-Based Ranger TI- 83 plus Calculator Overheard with TI-84 adapter Graph Paper Worksheet on CBR Paper Pencil Journals Day 4: Green Globs program Worksheet on writing equations Paper Pencil Journals Day 5: TI-83 plus Calculator Overhead with TI adaptor Class worksheet Find solutions worksheet Paper Pencil Journals 5-Day Overview

6 6 Day 1: Graphs of Linear Equations Class discussion on what makes up a line Creating a table of values of a linear equation Plotting points on Geometers Sketchpad to create a line Assessment: Hand in their class work from Geometer s Sketchpad and there journals. Students will need to complete a worksheet on graphing linear equations. Day 2: Using Intercepts Class discussion on: What is an intercept? Exploring the TI-83 Plus calculator Finding intercepts using the TI- 83 Plus calculator (looking at graph and table) Group work on finding intercepts Full class discussion on class work Assessment: Turn in journals and class work Day 3: Introduction to Slope Class discussion on positive, negative and zero slope Instruction on CBR Group work with CBR Instruction using slope formula Assessment: Worksheet on finding slopes of different lines and turn in journals Day 4: Writing an equation for a line Class Discussion of equations for a line Instructions on Green Globs program Individually work in Green Globs with prizes Assessment: Worksheet on writing equations and turn in journals Day 5: Introduction to Solving Systems of Equations Class discussion on solving systems of equations Instruction on solving systems of equations with TI- 83 Plus calculator Class worksheet on solving systems of equations Group work on solving systems of equations Assessment: Finding solutions to systems of equations worksheet and turn in journals Day One Lesson Plan Lesson Topic: Graphs of Linear Equations

7 7 Grade Level: 9 Materials: Geometer s Sketchpad, Worksheets, Computers, Journals Lesson Overview: Students will work with Geometer Sketchpad to develop an understanding of graphing linear equations.. Lesson Objectives: Students will demonstrate how to make a table of values. Students will be able to graph linear equations in Geometers Sketchpad. New York Standards: A.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equation, or objects created using technology as representation of mathematical concepts. A.R.3 Use representation as a tool for exploring and understanding mathematical ideas. A.G.4 Identify and graph linear functions A.A.21- Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable. NCTM Standards: Create and use representations to organize, record, and communicate mathematical ideas. Use Mathematical models to represent and understand quantitative relationships. Anticipatory Set: The teacher will open the class by asking a variety of different questions that pertain to graphing linear equations: 1. What makes up a line? 2. How many points make up a line? 3. If you are given a linear equation, what information would you need from the equation in order to graph it? Possible Responses: 1. A line consists of points 2. 2 points make a line 3. Points Developmental Activity: 1. The teacher will discuss with the students how to find the values of points on a linear equation by creating a table and solving for different values. The teacher will show the students an example of a linear equation and set up a table on the overhead. Then the teacher will show the students how to find the values, by algebraically solving for y when placing different numbers in for x, while using the overhead. (Worksheet)

8 8 2. The teacher will give the students a worksheet, with an example on how to find different values for y. Also on the worksheet will be many different linear equations for which the students will need to create a table and find a few points on that linear equation. 3. The teacher will walk the students through the procedure of graphing lines from the tables they have created using Geometers Sketchpad. Then show the students how to plot points, and then find the linear equation from those points within Geometers Sketchpad. 4. With the worksheet on linear equations the students will find a few points for each equation. Then each student will plot the points in Geometers Sketchpad and find the linear equation in the program. By finding the linear equation within Geometers Sketchpad, will be a way to check to see if they graphed the equation correctly. 5. Once the students find the linear equations in Geometers Sketchpad they will print there results. Closure: The students will write into their journals how to draw a linear equation on graph paper without using geometers sketchpad. The students will write down how they felt about Geometers Sketchpad and if they were confused about anything. Assessment: The teacher will collect the worksheet and printed results of the graphs they created in Geometers Sketchpad. The students will be given a handout which will include linear equations, which they will need to create a table of values and graph for homework. Worksheet Name: Date:

9 9 Given equation the equation y = 3x + 2 find at least 4 values for y by creating a table X Y Just by plugging -4, -2, 0, and 2 into the equation y = 3x + 2 for x, we were able to solve for y! Directions: Create a table of points for each linear equation. Then in Geometers Sketchpad plot the points on the graph and create the linear equation. 1. y = ½ x y = x 2 3. y = 6 4. y = x y = x y = 8x Homework Name: Date:

10 10 Directions: Similar to class, create a table of values for each linear equation. With your table of values, DRAW the linear equation on graph paper. 1. y = x y = -2x y = -7x y = 6 5. x = x + 3y = You are taking a photography class and need a digital camera. The payment plan for the camera can be modeled by the equation C = 10m +50, where C is the total amount paid and m is the number of months. Graph the equation and then estimate how much you pay in 12 months. Answer Key:

11

12 In 12 months you pay: \$170. Because when x = 12 the y-value is 170.

13 13 Day Two Lesson Plan Lesson Topic: Using Intercepts Grade Level: 9 Materials: TI-83 plus calculator, Overhead with calculator adaptor, Instruction Worksheet (graphing), Instruction Worksheet (finding intercepts), X-Y Intercepts Worksheet. Lesson Overview: Students will work with their calculators to develop an understanding of intercepts. Lesson Objectives: Students will be able to graph linear equations. Given linear equations, students will calculate the x and y intercepts on the graph. Given linear equations, students will be able to draw conclusions about linear equations and their intercepts. A.A.21- Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable. New York Standard: A.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design and explanation for the steps used in solving a problem. A.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid. A.R.3 Use representation as a tool for exploring and understanding mathematical ideas. NCTM Standards: Understand patterns, relations, and functions. Organize and consolidate their mathematical thinking through communication. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. Analyze and evaluate the mathematical thinking and strategies of others. Create and use representations to organize, record, and communicate mathematical ideas. Anticipatory Set: The teacher will begin class with a few questions: 1. What does intercept mean? 2. How many intercepts do you think there are in graphing a line? 3. What do you think intercepts represent in graphing a line? Possible Responses: 1. When something crosses something, where they cross is the intercept 2. 1 or 2 3. Where a line crosses the axis

15 15 (Graphing an equation) Graph the equation y = x First press the [Y=] button: 2. Now type your equation into the y1 slot: 3. Press the [Zoom] button: 4. Select Zoom Fit (It will graph your equation): You can play with the zoom features and the window button, to find an appropriate view Instruction

16 16 (Finding intercepts with a graph and table) Find the x-intercept and y- intercept of the equation y = x First graph the line: 2. To find x-intercept press [2 nd ] then [CALC] then select Zero. Make sure your curser is to the left of the x-axis and on the line and press [enter]. 3. Then move your curser so it passes the x-axis and still on the line, and select [enter]. 3. Now make a guess where you believe the line crosses the x-axis by moving your curser.

17 17 4. Finally the calculator gives us the point at which the line crosses the x-axis. The exact values that cross the x-axis. This line crosses the x-axis at point (-5, 0). 1. First graph the equation: Find the y-intercept of y = x Press [2 nd ] then press [CALC] and select Value: 3. Enter into 0 into the X= (because where the line crosses the y axis the value for x is zero. Then press [enter]. This point is where the line crosses the y-axis, which is (0, 5). This is the y-intercept point.

18 18 Find the x and y intercept point according to the table. 1. Press [2 nd ] then press [Tblset], In this situation, enter -10 for Tblstart and 1 for Tbl = 2. Press [2 nd ] then press [TABLE] 3. Now look for when x = 0 and where y = 0. These are your intercepts on the x and y axis. In this situation you can find that x-intercept = -5 and y-intercept = 5. X/Y Intercepts

20 20 1. y-intercept = -15 x-intercept = 3 2. y-intercept = 6.5 x-intercept = y-intercept = 3 x-intercept = y-intercept = 1 x-intercept = y-intercept = x-intercept = What kind of line has no y-intercept? A line that does not cross the y-axis, for example the equation x = 5 2. If the x-intercept of a line is positive and the y-intercept is negative, does the line slat up or down from left to right? Explain your reasoning. Slants up from left to right as the line would cross the y- axis through the negative portion of the axis and it would cross the x-axis through the positive portion of the x-axis. 3. What did you notice about the y-intercept and the equation? That the y-intercept matches the b in the linear equation y = mx + b. The slope is the coefficient of the x-value. Day Three Lesson Plan Lesson Topic: Introduction to Slope

21 21 Grade Level: 9 Materials: Calculator-Based Ranger (CBR 2), Overhead with TI-adapter, TI-83 plus Calculator, Graph paper, CBR Worksheet 1, Homework Worksheet, Journals, Pencil. Lesson Overview: Students will work the CBR to develop an understanding of slope. The students will also be introduced to the slope formula to compute slope. Lesson Objectives: Given instruction on the CBR, the students will create graphs and describe their slopes Given the slope formula, students will be able to find the slope of a line. Students will be able to describe slopes with appropriate mathematical vocabulary. Given the TI-83 calculators, students will be able to graph a linear equation and find the slope. New York State Standards: A.PS.4 Use multiple representations to represent and explain problem situations. A.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams. A.R.3 Use representation as a tool for exploring and understanding mathematical ideas. A.A.33 Determine the slope of a line, given the coordinates of two points on the line. A.A.32- Explain slope as a rate of change between dependant and independent variables. NCTM Standards: Understand patterns, relations, and functions. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. Use the language of mathematics to express mathematical ideas precisely. Create and use representations to organize, record, and communicate mathematical ideas. Anticipatory Set: The teacher asks the students to answer the following question: Determine whether each statement is positive slope, negative slope or no slope 1. John walks up a hill 2. Marie walks down a hill 3. Bill walks across flat ground. Possible Response:

22 22 1. Positive slope 2. Negative slope 3. No slope Developmental Activity: Lecture/Activities: 1. Teacher introduces the Calculator-Based Ranger (CBR 2) by direct instruction as it is connected to the overhead. The teacher does a few examples of walking close to the CBR and walking away from the CBR. Then shows the students the different graphs created by the CBR of him walking. The teacher will ask the students what do the graphs look like when I walk close to the CBR and what do the graphs look like when I walk away from the CBR? 2. The teacher discusses why the graphs look the way they do. For example, as the teacher walks close to the CBR the slope is negative. 3. To find slope the teacher traces the graphs that were created from the CBR and finds two random points. Then uses the slope formula to calculate the slope of the graphs. This will be done by direct instruction from the teacher on the board. 4. The students break up into groups of 3 and with their TI-83 plus calculators and CBR they need to complete a handout. (Worksheet 1) 5. The teacher will demonstrate to the students how to find the slope of a linear equation by finding the coefficient of the x term on the overhead, from a linear equation when the students have finished their CBR activity. Closure: 1. The students will write in there journals what they learned in class about different slopes. 2. The students will turn in there journals and their handout they completed with the CBR, before they leave class. Assessment: Give the student s homework on finding slopes of given points and there will also be a writing question for the students to do. (Worksheet 2) Worksheet 1 Name: Date:

24 24 1. Positive Slope. As I walked away the slope of the graph increased steadily, this is a positive slope graph: 3. Point 1: (0.0, 1.13) slope = =.50 Point 2: (2.7, 2.47) That the slope of the graph is.50. That I was walking away from the CBR at a rate of.5meters per second 5. Negative Slope. As I walked toward the CBR, the slope of the graph decreased steadily, this is a negative slope. 6. graph: Point 1: (0.0, 2.46) slope = = -.56 Point 2: (2.7,.96) The slope of the graph is That I was walking toward the CBR at a rate of.56 meters per second 9. Zero Slope. As I stood still the slope of the graph stayed the same. There is a zero slope.

25 slope Point 1: (.35,.85) Slope = = 0.0 Point 2: (.40,.85) There is no slope. The horizontal line gives you no slope. -8

26 26 Homework Name: Date: Directions: Find the slope of the line passing through the points. Also write down the type of slope of the points: (positive, negative, Zero or Undefined). HINT: If there is a zero in the numerator than the line is undefined. If there is a zero in the denominator the slope does not exist or undefined. Write a paragraph on the last question. Use your calculator, if needed. 1. (-4, 8), (6, 6) 2. (1, 4), (1, -7) 3. (-2, -4), (4, 2) 4. (-5, 4), (3,4) 5. (5, 8), (0, 5) 6. (-3, 1), (-3, -2) 7. (-6, -2), (6, -7) 8. (9, -8), (15, -8) 9. (12, 22), (-20, 19) 10. One line passes through eh points M (1, 1) and N (3, 4) and another line passes through points P (2, 5) and Q (5, 8). Which line has a greater slope? Explain why?

27 27 Answer Key: Undefined undefined The line that passes through points M and N has a slope of 1.5, which is a greater slope than the line that passes through points P and Q which has a slope of 1. The first line with point M and N has a greater number as a slope compared to the line with points P and Q.

28 28 Day Four Lesson Plan Lesson Topic: Writing an equation for a line Grade Level: 9 Materials: Computers, Green Globs Software, Homework handout, Pencil, Paper. Lesson Overview: Students will work with Green Globs software to develop an understanding of writing equations. Lesson Objectives: Given the slope-intercept formula, the students will be able to rewrite linear equations. Using the Green Globs software the students will be able to write linear equations. Given the slope and a point on the line, students will be able to write an equation of a line. New York State Standards: A.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts. A.A.34 Write the equation of a line, given its slope and the coordinates of a point on the line. A.G.4 Identify and graph linear functions. NCTM Standards: Represent and analyze mathematical situations and structures using algebraic symbols. Understand patterns, relations, and functions. Create and use representations to organize, record, and communicate mathematical ideas. Anticipatory Set: The teacher will ask the students to find the slope and y-intercept of a linear equation. 1. x y = y = 6 x 3. 1 = 2x -y Possible Response: 1. Slope = -1, y-intercept = Slope = -1, y-intercept = 6 3. Slope = 2, y-intercept = -1 Main Activity:

29 29 Lecture/Activities 1. The teacher will have a direct instruction on the overhead on the slopeintercept form of an equation: y = mx + b. 2. The direct instruction will have an example of rewriting an equation into the slope-intercept form, on the overhead. The example will be: 1 = 2x y which can be written as y = 2x The teacher will then review with the students by discussion, where the slope and y-intercept can be found in an equation. For example in the equation y = 2x 1, the slope is 2 and the y-intercept is The teacher will also show the students that if you are given a point and the slope you can write a linear equation. For example if you are given the point (1, 2) and slope = 2 then the equation is y = 2x + 2. This example will be done on the overhead. 5. The teacher will introduce the Green Globs program to the class. The teacher will show the students how to type equations into the program, by doing an example. 6. The students will work in the Linear and Quadratics Program. Within this program the students will work with lines. 7. Green Globs will give the students a line on a graph. The students are to write the equation of the line. If they are right they can advance to the next level. If they are wrong, the program will graph the line you typed and the student s can change your response so it is correct. 8. With this activity the students are learning by discovering. They are exploring linear equations through the Green Globs program. Closure: At the end of the class the students can play the Green Globs game. In this game the students try to hit all the green dots on the graph paper by writing equations and graphing them. If the line crosses through a green dot, the dot gets hit and you receive points. The more dots you hit with a line the more points you get. The top five students who get the most points will receive a cool mechanical pencil. Assessment: The students will first rewrite equations into the slope-intercept form in the homework assignment. Then students will also have to write down the equation of a line for each problem which will consist of a point on the line and the slope. (Homework Worksheet) Homework

30 30 Name: Date: Part I Directions: Write the equations into slope-intercept form 1. -4x + 2y = x 2y = y + 2 = 1/2x Part II Directions: Given a point on the line and the slope of the line write down the equation of the line. 1. (0, 2) Slope: 2 2. (1, 2) Slope: 1 3. (-4, 5) Slope: ½ 4. (3, 2) Slope: 5 5. (0, 10) Slope: (0, 0) Slope: 1 7. (-7, -8) Slope: (9, 10) Slope: (-15, -18) Slope: -2/3 10. (-5, -3) Slope: 1/5 Answer Key:

31 31 Part I 1. y = 2x y = 3x 5 3. y = 1/2x 2 Part II 1. y = 2x y = x y = 1/2x y = 5x y = -10x y = x 7. y = y = 11x y = -2/3x y = 1/5x 3 Day Five Lesson Plan

32 32 Lesson Topic: Introduction to system of equations Materials: TI-83 plus Calculator, Overhead with TI adaptor, Paper, Pencil Grade Level: 9 Lesson Overview: Students work with their TI-83 calculators to develop an understanding of solving a system of equations. Lesson Objectives: Given a technical briefing of the TI-83 calculator, students will be able to enter system of equations into the calculator. Given instruction on the matrix program of the TI-83 calculator, students will be able to solve systems of equation. Students will be able to explore equations on the calculator by graphing the linear equations. New York State Standards: A.PS.4 Use multiple representations to represent and explain problem situations. A.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, function, equations, charts, graphs, Venn diagrams, and other diagrams. A.R.3 Use representation as a tool for exploring and understanding mathematical ideas. A.G.9 Solve system of linear equations graphically. NCTM Standards Represent and analyze mathematical situations and structures using algebraic symbols. Apply and adapt a variety of appropriate strategies to solve problems. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. Create and use representations to organize, record, and communicate mathematical ideas. Anticipatory Set: The teacher will present the problem: Solve the linear system: x + 4y = 4 x- y = -6 The class will have a discussion with the teacher on what it means to solve a system of equation. Possible ideas: When the two linear lines cross. The two linear equations have the same x and y value. Developmental Activity:

33 33 Lecture/Activities: 1. The teacher will go through on the overhead how to use the matrix program in the TI-83 plus calculator. (Instructional Worksheet) 2. The students will follow with the teacher on how to solve a system equation in the matrix program as the teacher demonstrates with the calculator on the overhead. Each student will have a copy of the Instructional worksheet for a reference. 3. The teacher will also go through graphing the two linear equations on the calculator as a way to check their answers by finding the intersection. 4. The students will be given a worksheet of different system of equations, where they will need to work independently to solve the equations. (Worksheet) 5. When the students are finished with their worksheet they will get into groups of 3 and compare answers. 6. In their groups they will answer a few thinking questions that are on the handout. 7. When all the groups are finished discussing their answers and group questions, the class will come together and the teacher will have a class discussion on the problems they have been working on. Closure: The students will write into their journals about the lesson. They will write down something they learned and something they are still confused about in regards to the lesson. Assessment: The students will be given a homework assignment of many systems of equations. (Homework) Instructional Worksheet (System of Equations)

34 34 Solve the linear system: x + 4y = 4 x- y = -6 First make sure that the two equations are lined up by their variables. (Which they are) 1. Press [2 nd ] then press [MATRIX] 2. Move over from NAMES to EDIT using the arrow keys 3. Now press [ENTER] 4. Make sure that size of the matrix is the size that you want. (In this case it is 2 x 3, because there are two columns and 3 rows). If the system is larger or smaller change the numbers next to where it says MATRIX[A]. 5. Now move your curser into the matrix using the arrow keys and input the numbers of your system into the matrix.

35 35 6. Press [2 nd ], then press [Quit] once you have the matrix you want on the screen, this will take you back to the home screen. 7. Now Press [2 nd ], then press [CATALOG] 8. Now Press [r] and look for rref( and press [ENTER], rref( should appear on your home screen 9. Press [2 nd ], then press [MATRIX] 10. Make sure that that the number of the matrix that you imputed into your calculator is highlighted, in this case 1: [A] 2x 3, and press [ENTER]

36 Now press ) to close your equation and then press [ENTER] Your answer is located in the last column, in this problem the answer is (-4, 2) Class Work (Solving System of Equations)

38 38 2. (-4, -5) 3. (6, -3) Group Work Questions 1. This is the two lines cross. This is where the two linear equations have the same x and y values. 2. Yes, Solve for Y and substitute that equation into the other equation for Y. Now solve for x. This is your answer for x. Then put the x-value into an equation and solve for y. This is your y-value. 3. Yes, solve each equation for Y, and then graph both equations into your calculator. Go into CALC and find there intersection. Homework Name: Date: Directions: Solve the system of equations

39 39 1. x = 4 y = x a + b = 4 4a + b = w z = 2 4w + z = x + 4y = 14 3x -5y = You have 100 trading cards and your friend has 20, every day you give your friend one card. Use the equations c = 100 d and c = 20 + d to model this situation. Use the matrix or graph to find out when both of you have the same number of cards. Answer Key: 1. (4, 5)

40 40 2. (-1, 5) 3. (11/3, 16/3) or (3.67, 5.33) 3. (3, 2) 5. In 40 days they will have the same number of cards. They will both have 60 cards.

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