UNIT PLAN: SOLVING SYSTEMS OF EQUATIONS

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1 UNIT PLAN: SOLVING SYSTEMS OF EQUATIONS Course Algebra 1 Grade Level 9 th Time Span 6 Days, 80 minute periods Tools Algebra tiles, TI Graphing Calculators, Internet & class set of computers, projector for teacher computer, Alg Chapter 5 APP for TI graphing calculators, Green Globs BY: AMANDA STEPHENS FALL 2006

2 AUDIENCE: Algebra 1 TIME FRAME: minute period. NYS PERFORMANCE STANDARDS: AA7 Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables. AA10 Solve systems of two linear equations in two variables algebraically. AG7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variables. ACM10 Solve systems of linear equations using concrete models, graphs, tables, algebraic methods and check the reasonableness of solutions. NCTM STANDARDS: Understand vectors and matrices as systems that have some of the properties of the real-number system Develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper and pencil calculations for simple cases and technology for more complicated cases. Judge the reasonableness of numerical computations and their results. OBJECTIVES: By the end of this unit, students will be able to: 1. Solve systems of equations using various methods including substitution, elimination using addition and subtraction, and elimination using multiplication, 2. Use a graphing calculator to solve systems of equations by graphing and matrices. RESOURCES: Holliday, Cuevas, Moore-Harris, Carter, Marks, Casey, Day, Hayek, Algebra 1; New York State Edition, McGraw Hill, Glencoe, New York, Math A Regents Prep, Donna Roberts. Matrices Using the Graphing Calculator, Frederick & Donna Roberts. Equations.html. Solve Systems of Equations-Tutorial. Dr Abdelkader Dendane, Ph.D. MATERIALS: Graphing calculators Rulers Algebra tiles Markers Graph paper (incl) TI APP Algebra Ch. 5 Worksheets (incl) Glencoe, Algebra, NY Edition, Ch. 7 2

3 Quiz (incl) Smartview & Overhead Green Globs software Internet and Projector OVERVIEW DAY 1 Introduce idea of what is a system of equations Solve systems of equations by graphing Use graphing calculators & graph paper Key Vocab: system of equations, consistent, inconsistent, dependent, independent, intersection DAY 2 Solve systems by substitution Use Algebra Tiles Key Vocab: substitution DAY 3 Solve systems by elimination using addition or subtraction Key Vocab: Elimination DAY 4 Solve systems by elimination using multiplication DAY 5 Solve systems using matrices Use Graphing calculators DAY 6 Complete review/ take up review sheet Alg Ch 5 App from TI may be used to help review Quiz *NOTE: Days can be rearranged; only elimination by addition should be taught before elimination by multiplication. Use Algebra Ch 5 APP from TI at any point suitable. This app reviews consistent and independent, consistent and dependent, and inconsistent. It also explains the elimination method and has a couple of games to help students master the subject. ANTICIPATORY SET: Marcus went to the movies and bought 2 tubs of popcorn and 1 soda for $4.75. Jackie went to the movies and bought 1 tub of popcorn and 2 sodas for $4.25. What is the cost of 1 soda and 1 popcorn? DAY 1 GRAPHING Objectives Solving systems by graphing. ATTACHED: Graph paper, graphing activity, graphic organizer to help with terminology. Students should be familiar with graphing a line on the graphing calculator. If they are not, they need to know how to input a linear equation and how to read the table to plot the ordered pairs. 3

4 Key Terms In general, a solution of a system in two variables is an ordered pair that makes BOTH equations true. In other words, it is where the two graphs intersect, what they have in common. So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system. A consistent system is a system that has at least one solution. An inconsistent system is a system that has no solution; parallel lines. The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other equation. In other words, they end up being the same line. The equations of a system are independent if they do not share ALL solutions. They can have one point in common, just not all of them. Use green globs to show examples of the different types of systems and work on vocabulary. Graph the following systems then discuss the solutions and the terminology in the chart below: (*note these are not in y=mx+b format as Green Globs does not require this format decide whether this is a good time to work on solving for y or to work on the vocabulary; students need to be accustomed to seeing the equations in different forms and know that they can still solve them!) 1) x+2y=3; 3x-y=-5, 1 solution, consistent and dependent. 2) y=2x-3; 4x=2y+6, Infinite solutions, consistent and dependent. 3) 3x-y=-2; 3x-y=0, no solution, inconsistent; lines are parallel. Have students sketch the three different examples in their notebooks and label each with the appropriate vocabulary. The note should resemble the chart below. GRAPH INTERSECTING LINES SAME LINE PARALLEL LINES # OF SOLUTIONS One Solution Infinite Solutions No Solution TERMINOLOGY Consistent Independent Consistent Dependent Inconsistent Use the following websites as a reference or to teach students how to use the graphing calculator. - step by step instructions with visuals on solving systems using the graphing calculator. This is good because it shows the calculator screens. - step by step instructions with problems for students to work out using the graphing calculator. This site is nice because it leads right into hands on problems for students. 4

5 If the above websites are not used, the following should be completed. Using the graphing calculator, the equations must be in y=mx+b form. In Y= type in both equations. Hit GRAPH to show the lines. To find the point of intersection, the point needs to be on the viewing window. This may require adjusting the window or hitting zoom, fit. Then, find the intersection by 2 nd Trace (Calc), choose 5 Intersection. Follow the prompts on the screen, the calculator should move from curve to curve as enter is hit but it may not if it is close to the side of the window, it may need manually moved from curve to curve by using the right and left arrows. After choosing curve 1, curve 2, and guess, the point of intersection will be shown. The table (2 nd Graph) can also be used if the solution(s) is a whole number. The table can reset (2 nd window, change in TBL) the change in values to view decimal solutions, however, this gets tricky for a level 1 algebra course. The solution is where the x value has two of the same y values. For example if x is 2 and y1 is 5 and y2 is 5, the solution is (2, 5). Students should try to estimate the answer before going to the table as it will make the solution easier to locate. Use the smartview to show the students the steps on the overhead. Do examples that the window will need to be adjusted. If students have a difficult time adjusting the window, they can use zoom fit. Zoom standard is also a good key for them to remember as it brings back the standard viewing window. Ex. 1 Y=-6; 4x+y=2 - solve both equations for y - graph & find intersection (solution) - describe the solution Ans y=-6, y=-4x+2; (2, -6); one solution, consistent and independent. Ex. 2 X+y=24; 3x+y=44 - solve both equations for y - graph & find intersection (adjust window; xmax 15, ymax 50 or zoom fit) - describe the solution Ans y=-x+24, y=-3x+44; (10, 14); one solution, consistent and independent Pass out graph paper (attached below) and have students graph, find the solutions, and describe the solutions to the following systems. 1. 2x-y=-3; 8x-4y=-12; Ans - infinite solutions, consistent, dependent 2. x-2y=4; x-2y=-2; Ans -no solution, inconsistent 3. x+y=3; 12x+4y=20; Ans (1,2), one solution, consistent, independent 5

6 DAY 2 SUBSTITUTION Objectives Solve systems of equations using substitution. Introduce solving systems by substitution. Explain that substitution means replacing one variable so that there is only one variable to solve for instead of two. Give students an example of substitution in real life. If I have 2 coke and 1 pepsi, then replace the pepsi with a coke, now I have only coke. The main thing the students need to understand is that by substituting, they end up with only 1 variable. Model this method using algebra tiles. Use examples that solve for x first, that is, y is replaced. Ex. 1 3x+y=8, y=x-4; since y is x-4, replace y in the first equation with x-4. Using parenthesis whenever anything is replaced is helpful when students need to distribute so they have it set up correctly. In this problem, 3x+(x-4)=8. (Solution: (3,-1)) 3X { XX X X X X = X-4 { X -1-1 } Solve like a regular equation. Students can have the equation mats and also a worksheet where they solve the equation step by step. Use the value for x to solve for y by substituting x into either of the original equations. Ex. 2 4x+3y=10 and y=x+1 (Solution: (1,2)) Use the algebra tiles to solve this problem. Write each step out on paper. Sometimes, one equation is not solved in terms of the other variable. In these problems, first you must solve one equation so that it equals a variable. After one equation is solved, use the substitution method. Ex. 3 4x+y=12, -2x-3y=14 (Solution: (5, -8)) In this problem, since the y in equation 1 is equal to one, it is easier to solve this equation in terms of y; y=-4x+12. Substitute this new value for y into equation 2 where y is used. DON T forget the parenthesis!! -2x-3(-4x+12)=14. Solve for x. Use this value of x to solve for y. The website may be used to help teach this method or as a review after it is taught. - steps to solving equations using substitution Solve the following problems in student notebooks: 1. x=4y, 4x-y=75 (20, 5) 2. y=3x-8, y=4-x (3, 1) 3. x+3y=12, x-y=8 (9, 1) 6

7 4. x+14y=84, 2x-7y=-7 (14, 5) x+4y=-1, x+2.5y=3.5 (6, -1) 6. At the end of the 2000 baseball season, the NY Yankees and the Cincinnati Reds had won a total of 31 World Series. The Yankees had won 5.2 times as many World Series as the Reds. How many World Series did each team win? (Y+R=31, Y=5.2R; y-26, r-5). Take up these problems in class if time permits. Homework sheet is attached. DAY 3 ELIMINATION BY ADDITION/SUBTRACTION Objectives Solve systems of equations using elimination by addition and subtraction. Introduce solving by elimination using addition or subtraction. Elimination means getting rid of something, in this case, remove a variable. Again, the idea is to have only one variable to solve for. - steps to solving systems of equations using elimination. Given: x+y=62 Since the y values have a coefficients of positive & + x -y=12 negative one, add them together to eliminate y. 2x = x=37 Using x=37, substitute this new value for x into either of the original equations y = y =25 Therefore, the solution is x=37 and y=25. Given: a+b=15 Since the a values have coefficients of positive one, - a+2b=3 subtract to eliminate a. -1b= b=-12 Using b=-12, substitute this new value for b into either of the original equations. A+(-12)= a = 27 7

8 Students try: 1) a-w=120; a+w=150 Ans a=135, w=15 2) 4x+13y=40; 4x+3y=-40 Ans x=-16, y=8 3) A resort hotel offers two weekend specials; Plan A is 3 nights with 6 meals for $264, Plan B is 3 nights with 2 meals for $218. What is the cost of one night s lodging and what is the average cost per meal? Solution: 3N+6M=264-3N+2M=218 4M= M=11.5 The cost of a meal is $ Substitute m=11.5 into either original equation; 3N+6(11.5)=264 3N+69 = N = N =65 The cost for one night is $ Complete worksheet below in class or for homework. DAY 4 ELIMINATION BY MULTIPLICATION Objectives Solve systems of equations using elimination and multiplication. This method is similar to elimination by addition and subtraction, however one more step is added. If the coefficients of one of the variables are not the same, find a common multiple for one set and multiply each term by that number. Do the same for the second equation. In the end, one set of variables needs to have the same coefficient, either positive or negative. Students may want to multiply by a negative so they can add rather than subtract the two systems. Equations.html - a tutorial including step by step instructions on solving systems of equations by multiplication and elimination. Assignment: Glencoe Algebra 1 chapter 7-4, p.390, #1-12. DAY 5 MATRICES Objectives Solve systems of equations using matrices. The following website may be used to show students how a set of matrices can be used to solve systems of equations. 8

9 - steps to inputting matrices onto a calculator and how to solve the system using matrices. Students will want a copy of the steps to inputting a matrix system into their calculator and how to solve it. Steps to Using a Matrix to Solve Systems of Equations *Make sure the variables and numbers are lined up vertically underneath one another. 2m+3n=9 5m+4n=5 Matrix A is 2X Matrix B is 2X1 9 5 On the calculator, type 2 nd x -1, move over to edit, choose 1 or A. Input the data, ie 2 enter 2 enter 2 enter 3 enter 5 enter 4 enter. Input matrix 2. Type 2 nd x -1, move over to edit, choose 2 or B. Input the data; 2 enter 1 enter, 9 enter, 5 enter. Go back to home screen, 2 nd Mode (Quit). Type in the inverse of A multiplied by B. 2 nd x -1 1 or A, x -1 * 2 nd x -1 2 or B. The answer is in matrix form. The first variable is on top, m=-3 and the second variable is on the bottom, n=5. Try the following problems from the worksheet Solving Systems using Matrices using the steps above. As a final review, students need to be able to solve equations given any format. A work sheet is attached (Systems of Equations look for the tiger) that can be used in class or as homework. (No word problems) - review of all formats of systems of equations. (No word problems) ASSESSMENT See attached quiz 9

10 HMWK Solving Systems by Substitution Please solve the following systems of equations for x & y using the substitution method. Check your work. 1. y=3x x+2y= x=4y 2x+3y=44 3. x+5y=-3 3x-2y=8 4. c-5d=2 2c+d=4 5. 5r-s=5-4r+5s=17 10

11 SYSTEM OF EQUATIONS Solution Exists No Solution CONSISTENT INCONSISTENT One Solution Many Solutions INDEPENDENT DEPENDENT 11

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13 SYSTEM OF EQUATIONS Solution Exists No Solution One Solution Many Solutions 13

14 GRAPHING SYSTEMS OF EQUATIONS INVESTIGATION WORKSHEET Veronica is considering two job offers in car sales. The first job pays $400 per week plus 15% sales commission. The second job pays $350 per week plus 20% sales commission. What amount would Veronica have to sell for her weekly salary to be the same at either job? Problems: 1. Write two equations, one for each job, representing the amount of weekly sales, x, and total weekly salary, y. 2. Type these two equations into Y1 and Y2 in the graphing calculator. Graph the functions. What is on the screen? How does the viewing window need to be adjusted? 3. Find the point of intersection of the two lines. What is the ordered pair? 4. At what amount of sales will the two jobs have the same salary? What will that salary be? Solution 2. xmax 1200, ymax (1000, 550) 4. Sales $1000, Salary $550 14

15 Solving Systems of Equations Elimination by Addition & Subtraction Solve the following systems of equations using elimination by addition or subtraction. 1. 2x+8y =2-2x-4y=6 2. a+b=11 a-b=4 3. 2x-3y=5 5x-3y= m+n=-5 2m+3n=7 5. Suppose two eggs with bacon cost $2.70. One egg with bacon costs $1.80. At these rates, what will bacon alone cost? 6. Five gallons of regular unleaded gas and eight gallons of premium gas cost $ Five gallons of regular unleaded and two gallons of premium gas cost $8.75. Find the cost per gallon of each kind of gasoline. 15

16 Answers Solving Systems of Equations using Elimination by addition & subtraction. 1. x=-7 y=2 2. a=7.5 b= x=2 y=-1/3 4. m=-5.5 n=6 5. eggs=.90 bacon= regular = 1.19 premium=

17 Solving Systems of Equations Matrices Use the graphing calculator to solve the following systems of equations using matrices. Show your work and the key sequence of the calculator steps for full credit! 1. 2x+3y=15 5x-2y= x=4y+15 6y=3x-9 3. The sum of two numbers is 45. Three times the first number plus seven times the second is 115. Find the two numbers. 4. Tickets for a school play sell for $3 for a student and $5 for an adult. One night 595 people bought tickets. The school took in $1951. How many adult tickets and how many student tickets were sold? 5. Mary has 30 nickels and dimes totaling $2.40. How many of each does she have? 17

18 ANSWERS Systems of Equations using Matrices 1. x=-3, y=7 2. 5x-4y=15, -3x+6y=-9; x=3, y=0 3. x+y=45, 3x+7y=115; x=50, y=-5 4. s+a=595, 3s+5a=1951; s=512, a=83 5. n+d=30, 5n+10d=240; n=12, d=18 18

19 Systems of Equations Quiz 1. Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. Use substitution to solve the system of equations. 2. Use elimination to solve the system of equations At a local electronics store, CDs were on sale. Some were priced at $14.00 and some at $ Sabrina bought 9 CDs and spent a total of $ How many $12.00 CDs did she purchase? 19

20 Systems of Equations Answer Section SHORT ANSWER 1. ANS: one solution; ( 3, 5) Graph each line. The point where the two lines intersect is the solution. Check the solution by replacing x and y in the original equations with the values in the ordered pair. DIF: Average OBJ: Solve systems of equations by graphing. STO: A.A.7, A.G.7 TOP: Solve systems of equations by graphing KEY: System of Equations, Graphing MSC: 1998 Lesson 8-1 NOT: /A/ Remember that the x-coordinate comes first in an ordered pair. /B/ Did you graph the second line correctly? /C/ Graph both lines./d/ Correct! 2. ANS: (33, 36) Substitute the right side of the top equation for y in the bottom equation. Solve for x. Substitute the value obtained for x into the top equation and solve for y. DIF: Average OBJ: Solve systems of equations by using substitution. STO: A.A.7, A.A.10, A.G.7 TOP: Solve systems of equations by using substitution KEY: System of Equations, Substitution MSC: 1998 Lesson 8-2 NOT: /A/ Remember to list the x-coordinate first and then the y-coordinate./b/ Use parentheses when you sustitute for y and then remember to distribute the negative sign through the parentheses. /C/ Terms with different variables cannot be combined. /D/ Correct! 3. ANS: ( 1, 1) 20

21 Eliminate one variable by adding the two equations. Solve for x and then substitute that value into one of the equations to find the value of y. DIF: Average OBJ: Solve systems of equations by using elimination with addition. STO: A.A.7, A.A.10, A.G.7 TOP: Solve systems of equations by using elimination with addition KEY: System of Equations, Elimination, Addition MSC: 1998 Lesson 8-3 NOT: /A/ Correct! /B/ Double-check your positive and negative signs. /C/ Add the equations together./d/ Add the equations together. 4. ANS: 6 Solve the first equation for one of the variables and substitute into the second equation. Solve. Substitute that value into the first equation to find the second value. DIF: Average OBJ: Solve real-world problems involving systems of equations. STO: A.A.7, A.A.10, A.G.7 TOP: Solve real-world problems involving systems of equations KEY: System of Equations, Real-World Problems MSC: 1998 Lesson 8-2 NOT: /A/ This is the total number of CDs purchased. /B/ Correct! /C/ Does your answer satisfy the equations? /D/ This is the number of $14.00 CDs purchased. 21

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