Table of Contents



Unit 3Systems of Linear Equations and InequalitiesTest Date: ___________________11950705524500Name: ___________________________________________________________________________________By the end of this unit, you will be able to…Graph a system of linear equations and identify the solution from the graphSolve rigorous systems of linear equations by an appropriate method (graphing, substitution, or elimination)Graph a system of linear inequalities and identify the solution set from the graphCreate a two-variable system of linear equations to represent an application problemSolve application problems of two variables using a system of linear equationsIdentify if a solution is true for a system of equations given the graph, table, or equationsTable of Contents TOC \o "1-3" \h \z Graphing Systems of Linear Equations PAGEREF _Toc494362573 \h 4Substitution PAGEREF _Toc494362574 \h 9Elimination PAGEREF _Toc494362575 \h 13Graphing Calculator Lab PAGEREF _Toc494362576 \h 17Graphing Systems of Linear Inequalities PAGEREF _Toc494362577 \h 18Introduction – Group ChallengeA father challenged his daughter to a race. He gave his daughter a 40 meter head start. He ran at 9 meters per second, and his daughter ran at 4 m per second.Will he catch his daughter in a 100 meter race? If so, when will he catch her and how far will they be from the starting line?If not, determine how far he would have to go to catch her.Show all of your work. I have provided tables and a graph as a resource, but you can use any method.39433527940Graphing Systems of Linear Equations485013047307500Warm Up: It’s a race! You and your friend are about to run a race, but your friend cheated and stopped to grab their bike. The equations below represent y, your distance in miles from the start, in terms of x, the number of minutes after the race begins. You: y=16x Your friend: y=13x-1Graph the two lines on the same coordinate plane.How many minutes after the start does your friend pass you?System of Equations: ___________________________________________________________.Solution to a System: The __________________________________ that is a ______________________ to both equations.For example, the solution to the system y=16xy=13x-1 is __________________.Possible Number of SolutionsA system of equations could have:One solutionNo solutionInfinitely many solutionsDirections: Graph the system of equations. Then determine whether the system has one solution, no solutions, or infinitely many solutions. If the system has one solution, name it.Example 1y=16xy=13x-1One solutionSolution: (6, 1)(see Warm Up)Example 2y=2x+3 8x-4y=-12Example 3x-2y=4 x-2y=-2-69215120650004966335-22606000Example 4: Naresh rode 20 miles on his bike last week and plans to ride 35 miles per week. Diego rode 50 miles last week and plans to ride 25 miles per week. Predict the week in which Naresh and Diego will have ridden the same number of miles.Define your variables.x= _____________________________________________y= ______________________________________________Write an equation for Naresh and an equation for Diego.y= ______________________ y= _______________________Graph to solve the system.48520355143500Example 5: Alex and Amber are both saving money for a summer vacation. Alex has already saved $100 and plans to save $25 per week until the trip. Amber has $25 and plans to save $50 per week. In how many weeks will Alex and Amber have the same amount of money?Summarize what we’ve learned:# of solutions: ____________________ # of solutions: _________________# of solutions: _______________________How do you find the solution to a system of equations? ___________________________________________________________________________________________________________________________________________________________________________________Solving by GraphingGraph each system and determine the number of solutions that it has. If it has one solution, name it.1. 2x – y = 1 2. x = 1 3. 3x + y = –3y = –3 2x + y = 4 3x + y = 3163830469905151755469902840309472694. y = x + 2 5. x + 3y = –3 6. y – x = –1x – y = –2 x – 3y = –3 x + y = 318631880366286261165498521923680366Freshman Class ButtonsPayton has an idea that could raise money for the freshman class. She would like to sell buttons commemorating her graduating class.Selling Price of Each ButtonNumber of Buttons in Stock (Supply)Number of Buttons that Students will Buy (Demand)$1.0035530$2.00130400$4.0032014050863517145000Plot points representing supply for each price in the table. Draw the line through the data points and write Supply on this line.Plot points representing the number of buttons requested (demand) for each selling price on the same graph. Draw the line through these points. Label this line Demand.Adapted from NCTM Illuminations ? National Council of Teacher of Mathematics Your GraphIf Payton sets the price at $2.50 per button, how many disappointed customers can she expect to have? Show or tell how you got your answer.If Payton sets the price at $3.80 per button, how many unsold buttons can she expect to have left over? Show or tell how you got your answer.If Payton gives the buttons away at no charge, how many buttons would she need? How does the graph help you determine your answer?What price would make the button supply so low that the number of available buttons would be zero? How does the graph help you determine your answer?Estimate the price at which supply and demand will be in equilibrium. What is this price and how many buttons can Payton expect to sell? How does the graph help you determine your answer?Using EquationsUse your graph to find the equation for supply (S) as a function of price (P). Use your graph to find the equation for demand (D) as a function of price (P).Solve the system of supply-and-demand equations to find the price and the number of buttons that Payton should order for supply and demand to be in exact equilibrium. How does this price compare with your answer in question 7?4509135381000Write the steps in your own words:____________________________________________________________________________________________________________________________________0Write the steps in your own words:____________________________________________________________________________________________________________________________________SubstitutionStep 1: When necessary, solve at least one equation for one variable.Step 2: Substitute the resulting expression from Step 1 into the other equation to replace the variable. Then, solve the equation.Step 3: Substitute the value from Step 2 into either equation, and solve for the other variable. Write the solution as an ordered pair.Step 4: Check your answer by testing your ordered pair in one or both of the original equations.-63500179070We will be “substituting” “players” out between the two equations, like in a soccer match.But we can only substitute out equivalent players.(I promise, this will be the only sports analogy I ever make.)See below. Since -4x+12 is the same thing as y (they are equal), we can circle it and “substitute” it in for y in the second equation.1422400293370Example 1: Solve the system y=-4x+122x+y=2y=-4x+122x+y=23137535217805002x+-4x+12=2 Substitution2x-4x+12=2 Removed parentheses.-2x+12=2 Combined like terms.-2x=-10 Subtracted 12 from both sides.x=5 Divide by -2.1583055-2393950y=-45+12 Substitutiony=-20+12 Multiplicationy=-8 Combined like termsCheck in 1st Equation: -8=-45+12-8=-20+12 Solution: __________________Example 2 – You Try: Solve the system y=2x3x+4y=11y=2x3x+4y=11Example 3 – You Try: Solve the system y=4x+1y=2x+2y=4x+1y=2x+2Sometimes, you need to solve for a variable first. If you can, choose one that already has a coefficient of 1.Example 4: Solve the system x-2y=-33x+5y=24Use this space to solve one equation for one variable:Example 5 – You Try: Solve the system 3x-y=-12-4x+2y=20Use this space to solve one equation for one variable:Draw your own T-chart and solve the system using substitution:Example 6: A nature center charges $35.25 for a yearly membership and $6.25 for a single admission. Last week it sold a combined total of 50 yearly memberships and single admissions for $660.50. How many memberships and how many single admissions were sold?Example 7: A chicken farmer also has some cows for a total of 30 animals. The animals have 74 legs in all. How many chickens does the farmer have?Substitution - Special Cases3667125-10795Summarize:00Summarize:Solve the system y=2x-4-6x+3y=-12Solve the system 2x+2y=8x+y=-2Solve the system 2x-y=8y=2x-3Solve the system 4x-3y=16y-8x=-2-62865459740Warm Up: Two apples and three oranges cost $6. Write an equation that represents this scenario.Five apples and seven oranges cost $18. Write an equation that represents this scenario.How much would seven apples and ten oranges cost all together?0Warm Up: Two apples and three oranges cost $6. Write an equation that represents this scenario.Five apples and seven oranges cost $18. Write an equation that represents this scenario.How much would seven apples and ten oranges cost all together?EliminationImportant Idea: ______________________________________________________________________________________________________4800600340360Steps:_____________________________________________________________________________________________________________________________0Steps:_____________________________________________________________________________________________________________________________Sometimes it is easier to use a different method to solve systems of equations. The elimination method uses the following steps:When necessary, multiply one whole equation by a constant to make sure terms will cancel. When necessary, rearrange.Add the equations together. One variable should cancel out.Solve for the remaining variable.Substitute your solution from step 3 back into either equation and solve for the other variable.Check your answer by using the other equation. You should get the same result.4914900224790Step 500Step 5Example 1: Solve the system using the elimination method 4x+6y=323x-6y=3274320047625Step 400Step 4114300219710Step 2+0Step 2+5029200105410Check: 35-6y=315-6y=3-15 -15-6y=-12÷2 ÷2y=2 Check: 35-6y=315-6y=3-15 -15-6y=-12÷2 ÷2y=2 49149001054100285750010541045+6y=3220+6y=32-20 -206y=12÷6 ÷6y=245+6y=3220+6y=32-20 -206y=12÷6 ÷6y=22743200105410002286002197104x+6y=323x-6y=37x=35÷7 ÷7x=54x+6y=323x-6y=37x=35÷7 ÷7x=5114300733425Step 3Step 3Example 2: Solve the system using the elimination method -4x+3y=-34x-5y=5.Example 3: Use elimination to solve the system 5r+2t=69r+2t=22Example 4: Use elimination to solve 8b+3c=118b+7c=7Example 5: Use elimination to solve 5x+6y=-82x+3y=-5Example 6: Use elimination to solve 4x+y=123x+2y=19Solving Systems of Equations – Which Strategy Works Best?Discuss in your group: Which method did you choose for each problem? Why? Do you think the method you chose was the best choice? You DO NOT need to solve the systems!In your group, decide which method is best for each problem. Write the problem number in the appropriate column. Be prepared to explain why you selected that strategy to solve the problem, including the characteristics of the system. GraphingSubstitutionElimination2x-y=8x+y=42. 2t+3n=95t-3n=53. 2x-5y=73x-2y=-174. 5a-b=-94a+3b=-115. 2r+s=11r-s=26. y=3-x5x+3y=-17a+6b=015a-6b=08. y=3x-1y=2x-59. 2y=8-7x4y=16-14xSummarize-11430035623500: When is it best to solve a system by graphing, substitution, or elimination?Use your Venn diagram to identify the best method to solve the following problems. Explain your reasoning.2219960111125Best method: 00Best method: 2x-y=47x+3y=27 Why?2181225113030Best method: 00Best method: 4x+2y=-145x+3y=-17 Why?2181225127635Best method: 00Best method: 2x+7y=1x+5y=2 Why?Graphing Calculator LabYou can solve a system of equations using your graphing calculator.Example 1:5.23x+y=7.486.42x-y=2.11Solve each equation for y.y = __________________________y = __________________________503872514541500Enter these equations into the Y= list and graph in the standard viewing window (Zoom + 6). Sketch what you see.Press [2nd] + [CALC] + [5] + [ENTER] + [ENTER] + [ENTER]. Write what you see on the bottom of your screen on the line below._______________________________________________________________What does this tell you?504253516510000Example 2: Enter the following equations into the Y= list: y=-9x+100y=20x-1056Graph in the standard viewing window (Zoom + 6). Sketch what you see. Press [Window]. Enter the following information:Xmin=-100Xmax=100Xscl=10Ymin=-2000Ymax=200024701546355000Yscl=100Press [Graph]. Sketch what you see.Explain what you did in step 3 and why it changed the image.Press [2nd] + [CALC] + [5] + [ENTER] + [ENTER] + [ENTER]. Write what you see on the bottom of your screen on the line below._______________________________________________________________What does this tell you?Graphing Systems of Linear Inequalities4676775145415Steps:____________________________________________________________________________________________________0Steps:____________________________________________________________________________________________________Solving a System of Linear Inequalities by GraphingStep 1: Solve both of the inequalities for y (slope-intercept form)Step 2: Graph the first inequality and shade the appropriate region.Step 3: Graph the second inequality on the same coordinate plane and shade the appropriate region.Step 4: The solutions of the system are represented by: ________________________________________-114300128905Tips (feel free to add your own):Use a solid line for ≤ or ≥ . Use a dashed line for < or >.Use a test point to decide which side to shade.44577005016500Example 1: y<x+2y≥-52x+5TP1: TP2: What is one ordered pair that is part of your solution set? __________445770010668000Example 2: y≥23x+1y<4x-3TP1: TP2:What is one ordered pair that is part of your solution set? __________Example 3: y≤3x+y>54457700-45910500TP1:TP2:What is one ordered pair that is part of your solution set? __________445770015430500Example 4: y>12x+42y≥-6x-2TP1:TP2:What is one ordered pair that is part of your solution set? __________Example 5: A college service organization requires that its members maintain at least a 3.0 grade point average and volunteer at least 10 hours a week. Define the variables and write a system of inequalities to represent this situation. Then, graph the system.45091357937500Variables: x = ______________________ y = ______________________System of inequalities:What is one ordered pair that is part of your solution set? ____________ ................
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