UNIT 6: SYSTEMS OF LINEAR EQUATIONS - Weebly

UNIT 6: SYSTEMS OF LINEAR EQUATIONSTopic 1: Graphing and Substitution WHY LEARN THIS?Equations can be used in many real-worldapplications. For example, a hydrologist mightuse an equation to determine the differencebetween two rivers’ discharge rates. Equationsare also used with many businesses.Lesson 1: Solving Systems by Graphing(pg. 5)Lesson 2: More Practice with SolvingSystems by Graphing (pg. 13)Lesson 3: Solving Systems by Substitution(pg. 17)Lesson 4: Solving Systems by SubstitutionUsing Isolation (pg. 21)CAREER: BUSINESS AND TOURISMBusiness decision makers often use systems oflinear equations to model a real-world situationin order to predict future events. Being able tomake an accurate prediction helps them plan andmanage their businesses.Topic 2: Elimination Lesson 1: Solving Systems by Elimination(pg. 27) Lesson 2: Solving Systems by Elimination:Different Coefficients (pg. 31) Lesson 3: Solving Systems Using AllMethods (pg. 35)Trends in the travel industry change with time.For example, in recent years, the number oftourists traveling to South America, theCaribbean, and the Middle East is on the rise.Topic 3: Applications Lesson 1: System of EquationsApplications (pg. 41) Lesson 2: More Practice with Systems ofEquations Applications (pg. 49)1

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TOPIC 1:GRAPHINGANDSUBSTITUTION3

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Unit 6 Topic 1 Lesson 18th Grade MathDate:FOCUS QUESTION: How do you solve a system of equations?OBJECTIVES:VOCABULARY: I can solve a system of equations graphically. System of EquationsSolutionSolving Systems of Equations by GraphingSYSTEM OFEQUATIONSTheSOLUTIONto aSystemTYPES OFSOLUTIONSEXAMPLESGraphically: The point (x, y) where the two lines .Algebraically: The point (x, y) that makes both equations .INTERSECTING LINESPARALLEL LINESSAME LINEONE SOLUTIONNO SOLUTIONINFINITE SOLUTIONSDirections: Write the system of equations and identify the solution.1)System of Equations:Solution:5

2)System of Equations:Solution:3)System of Equations:Solution:4)System of Equations:Solution:5)System of Equations:Solution:6)System of Equations:Solution:6

Solve aSystem byGRAPHINGExamplesDirections: Solve the system of equations by graphing.5𝑦 4𝑥 1𝑦 2𝑥 81) {𝑦 2𝑥 52) {1𝑦 6𝑥 8Solution:Solution:𝑦 𝑥 53) {𝑦 3𝑥 73𝑦 4𝑥 84) {5𝑦 𝑥 84Solution:Solution:7

1𝑦 3𝑥 25) {𝑥 9𝑦 2𝑥 56) {𝑦 2𝑥Solution:Solution:𝑦 2𝑥 27) {5𝑦 𝑥 7𝑦 𝑥 528) {1𝑦 4𝑥 2Solution:Solution:128

2𝑦 𝑥 6𝑦 𝑥 6𝑦 3𝑥 19) {𝑦 310) {Solution:Solution:9

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Name:Unit 6 Topic 1 Homework 1: Solving Systems by GraphingDate:Directions: Write the system of equations shown on the graph and identify its solution.1)2)Solution:Solution:Directions: Solve each system by graphing. Be sure to clearly give the solution.2𝑦 𝑥 4𝑦 𝑥 33) {34){𝑦 5𝑥 2𝑦 𝑥 71𝑦 2𝑥 45) {5𝑦 2𝑥 2𝑦 2𝑥 76) {𝑦 511

7) {9) {𝑦 𝑥 4𝑦 𝑥 4𝑦 𝑥 7𝑦 4𝑥 3𝑦 2𝑥 411) {𝑦 2𝑥 18) {12𝑦 𝑥 23𝑦 4𝑥 33𝑦 2𝑥 910) {1𝑦 4𝑥 13𝑦 𝑥 2412) {𝑥 412

Unit 6 Topic 1 Lesson 28th Grade MathDate:FOCUS QUESTION: How do you solve a system of equations?OBJECTIVES:VOCABULARY: I can solve a system of equations graphically. System of EquationsSolutionSolving Systems of Equations by GraphingSYSTEM OFEQUATIONSThe SOLUTIONto a SystemTYPES OFSOLUTIONSTwo or more equations with the same variables.Graphically: The point (x,y) where the two lines intersect.Algebraically: The point (x,y) that makes both equations true.INTERSECTING LINESPARALLEL LINESSAME LINEONE SOLUTIONNO SOLUTIONINFINITE SOLUTIONS(x, y) SOLVING(1) Rewrite the equations in slope-intercept form.SYSTEMS BY(2) Graph the line.GRAPHING(3) Identify the solution(s).Directions: Solve each system of equations by graphing.1𝑦 𝑥 8𝑦 2𝑥 91){2){𝑦 2𝑥 1𝑦 𝑥 613

3){𝑦 3𝑥 8𝑦 𝑥 215){𝑦 𝑥 53𝑦 77){𝑦 5𝑥 7𝑦 2𝑥 6314) {6) {𝑦 2𝑥 212𝑦 𝑥 2𝑦 𝑥 5𝑦 𝑥 2𝑥 28) {𝑦 3 𝑥 9214

Name:Date:Unit 6 Topic 1 Homework 2: More Practice with Solving Systems by GraphingDirections: Solve each system of equations by graphing. Clearly identify your solution.2𝑦 𝑥 7𝑦 𝑥 12) {31){𝑦 𝑥 4𝑦 𝑥 4𝑦 𝑥 13){𝑦 1 𝑥4 415){𝑦 3𝑥 51𝑦 3𝑥 15𝑦 2𝑥 24) {9𝑦 2𝑥 65𝑦 4𝑥 36) {3𝑦 4𝑥 515

7){𝑦 𝑥 1𝑦 5𝑦 2𝑥 18) {3𝑦 4 𝑥 103𝑦 2𝑥 810) {𝑥 59){𝑦 2𝑥 8𝑦 𝑥 7Questions:1) If a system of equations has one solution, what does this mean about the two lines?2) If a system of equations has no solution, what does this mean about the two lines?3) If a system of equations has infinitely many solutions, what does this mean about the two lines?16

Unit 6 Topic 1 Lesson 38th Grade MathDate:FOCUS QUESTION: How do you solve a system of equations?OBJECTIVES:VOCABULARY:System of Equations I can solve a system of equations usingsubstitution.SolutionSolving Systems of Equations by SubstitutionWARM-UPDirections: Solve the following equations.1) 3x 16 – 52) x – 27 7x – 153) 6x – (2x – 9) – 314) 2x 8(3x 3) 3(x – 13)SolvingSystems bySUBSTITUTIONSteps toSolveExamples(1)(2)(3)(4)(5)Look for an equation that is solved for x or y.Substitute the expression into the other equation for that variable.Solve! Now you have one variable.Substitute your answer into either equation to find the other variable.Write your answer as an ordered pair (x, y).𝑦 𝑥 91) {𝑦 4𝑥 24𝑦 8𝑥 142) {𝑦 5𝑥 1117

5𝑥 4𝑦 73) {𝑦 2𝑥 84) {𝑦 4𝑥 23 4𝑥 3𝑦 135) {𝑦 43𝑥 5𝑦 236) {7𝑥 𝑦 23𝑦 4𝑥 114𝑥 9𝑦 487) {𝑥 38) { 𝑥 𝑦 1𝑦 𝑥 49) {𝑦 𝑥 82𝑥 2𝑦 1618

Name:Unit 6 Topic 1 Homework 3: Solving Systems by SubstitutionDate:Directions: Solve each system of equations by substitution.𝑦 4𝑥 13𝑦 3𝑥 201){2) {𝑦 6𝑥 19𝑦 2𝑥 12𝑦 5𝑥 193) {𝑦 5𝑥 15) {𝑦 4𝑥 29𝑥 𝑦 14) {𝑦 4𝑦 2𝑥 20𝑦 10𝑥 26) { 5𝑥 4𝑦 2719

6𝑥 𝑦 237) {𝑦 4𝑥 79) {𝑥 𝑦 10𝑦 2𝑥 711) {12𝑥 3𝑦 21𝑦 4𝑥 78) {𝑦 6𝑥 19 𝑥 2𝑦 12𝑥 210) { 5𝑥 4𝑦 10𝑦 5𝑥 412) { 10𝑥 2𝑦 1220

Unit 6 Topic 1 Lesson 48th Grade MathDate:FOCUS QUESTION: How do you solve a system of equations?OBJECTIVES:VOCABULARY:System of Equations I can solve a system of equations usingIsolationsubstitution.SolutionSolving Systems of Equations by SubstitutionWARM-UP1) Solve for y: 2x y 5Substitutionthat RequiresISOLATIONThe first step to solve a system of equations by substitution is to look for an equation that issolved for x and y. If there is not an equation, you will need to create one!Examples2) Solve for x: x – 4y 7Directions: Solve each system of equations using substitution. 3𝑥 𝑦 81) {8𝑥 3𝑦 19 𝑥 2𝑦 162) {4𝑥 𝑦 13) {2𝑥 4𝑦 25𝑥 𝑦 63𝑥 𝑦 144) {9𝑥 3𝑦 4221

𝑥 2𝑦 185) {8𝑥 7𝑦 176) {3𝑥 2𝑦 20𝑥 7𝑦 227) {3𝑥 12𝑦 6𝑥 4𝑦 88) {𝑥 4𝑦 13𝑥 9𝑦 269) {8𝑥 7𝑦 21𝑥 𝑦 12𝑥 𝑦 1110) {𝑥 4𝑦 2922

Name:Date:Unit 6 Topic 1 Homework 4: Solving Systems by Substitution Using IsolationDirections: Solve each system of equations by substitution.4𝑥 𝑦 13𝑥 4𝑦 241){2) {5𝑥 2𝑦 155𝑥 𝑦 239𝑥 2𝑦 143) { 7𝑥 𝑦 165) { 2𝑥 𝑦 14𝑥 2𝑦 24) {3𝑥 𝑦 272𝑥 3𝑦 4𝑥 𝑦 106) {4𝑥 3𝑦 1223

𝑥 5𝑦 217) {3𝑥 4𝑦 199) {𝑥 2𝑦 25𝑥 6𝑦 2211) {𝑥 𝑦 73𝑥 7𝑦 298) {5𝑥 2𝑦 8𝑥 3𝑦 14𝑥 4𝑦 1510) {3𝑥 12𝑦 12𝑥 2𝑦 2212) {2𝑥 𝑦 924

TOPIC 2:ELIMINATION25

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Unit 6 Topic 2 Lesson 18th Grade MathDate:FOCUS QUESTION: How do you solve a system of equations?OBJECTIVES:VOCABULARY:System of Equations I can solve a system of equations usingCoefficientelimination.SolutionSolving Systems of Equations by EliminationSolvingSystems )Make sure the equations are lined up!Identify a variable that has the same coefficient on both equations.Subtract the equations to eliminate this variable.Solve the equation for the remaining variable.Substitute your answer into either equation to find the other variable.Write your answer as an ordered pair (x, y).𝑦 9𝑥 231) {𝑦 𝑥 12) {𝑦 7𝑥 5𝑦 4𝑥 63) {𝑦 5𝑥 29𝑦 7𝑥 194) {𝑦 3𝑥 15𝑦 2𝑥 827

5) {𝑥 𝑦 3𝑥 𝑦 76) {7𝑥 4𝑦 117𝑥 10𝑦 257) {4𝑥 2𝑦 30 𝑥 2𝑦 58) {5𝑥 2𝑦 254𝑥 2𝑦 189) {4𝑥 𝑦 117𝑥 𝑦 263𝑥 𝑦 910) {3𝑥 𝑦 428

Name:Unit 6 Topic 2 Homework 1: Solving Systems by EliminationDate:Directions: Solve each system of equations by elimination.1) {𝑦 3𝑥 13𝑦 7𝑥 172) {𝑦 4𝑥 15𝑦 2𝑥 33) {𝑦 4𝑥 14𝑦 10𝑥 284) {𝑦 7𝑥 29𝑦 3𝑥 96) {2𝑥 9 172𝑥 3𝑦 195) {𝑥 𝑦 13𝑥 𝑦 529

7) {3𝑥 𝑦 73𝑥 𝑦 78) {9) {2𝑥 𝑦 57𝑥 𝑦 2010) {5𝑥 𝑦 263𝑥 𝑦 1812) {11) { 8𝑥 𝑦 6 8𝑥 3𝑦 14 4𝑥 5𝑦 122𝑥 5𝑦 63𝑥 2𝑦 53𝑥 2𝑦 1630

Unit 6 Topic 2 Lesson 28th Grade MathDate:FOCUS QUESTION: How do you solve a system of equations?OBJECTIVES:VOCABULARY:System of Equations I can solve a system of equations usingCoefficientelimination.Solving Systems of Equations by EliminationWhat if thereare NOCOMMONCOEFFICIENTS?EXAMPLES1) {𝑥 3𝑦 24𝑥 7𝑦 272) {7𝑥 2𝑦 32𝑥 5𝑦 473) {2𝑥 5𝑦 394𝑥 3𝑦 434) {4𝑥 𝑦 163𝑥 2𝑦 1731Solution

5) { 3𝑥 5𝑦 242𝑥 𝑦 166) {2𝑥 3𝑦 207𝑥 𝑦 137) {2𝑥 3𝑦 15𝑥 9𝑦 28) {2𝑥 5𝑦 286𝑥 𝑦 289) {2𝑥 4𝑦 9𝑥 2𝑦 1532

Name:Date:Unit 6 Topic 2 Homework 2: Solving Systems by Elimination – Different CoefficientsDirections: Solve each system of equations by elimination.1) {𝑦 3𝑥 13𝑦 7𝑥 172) {𝑦 4𝑥 15𝑦 2𝑥 33) {𝑦 4𝑥 14𝑦 10𝑥 284) {𝑦 7𝑥 29𝑦 3𝑥 96) {2𝑥 9 172𝑥 3𝑦 195) {𝑥 𝑦 13𝑥 𝑦 533

7) {3𝑥 𝑦 73𝑥 𝑦 78) {9) {2𝑥 𝑦 57𝑥 𝑦 2010) {5𝑥 𝑦 263𝑥 𝑦 1812) {11) { 8𝑥 𝑦 6 8𝑥 3𝑦 14 4𝑥 5𝑦 122𝑥 5𝑦 63𝑥 2𝑦 53𝑥 2𝑦 1634

Unit 6 Topic 2 Lesson 38th Grade MathDate:FOCUS QUESTION: How do you solve a system of equations?OBJECTIVES:VOCABULARY:System of Equations I can solve a system of equations usingIsolationgraphing, substitution, and elimination.SYSTEMS OF EQUATIONS REVIEWSolve each system of equations by GRAPHING. Clearly identify your solution.1𝑦 𝑥 3𝑦 𝑥 322) {1) {2𝑦 3𝑥 7𝑥 4Solve each system of equations by SUBSTITUTION. Clearly identify your solutions.𝑦 7𝑥 6𝑥 7𝑦 213) {4) {4𝑥 3𝑦 162𝑥 14𝑦 425) {3𝑥 5𝑦 15𝑥 4𝑦 126) {35𝑦 58𝑥 5𝑦 17SolutionCoefficient

Solve each system of equations by ELIMINATION. Clearly identify your solution.𝑥 𝑦 102𝑥 2𝑦 288) {7) {8𝑥 2𝑦 22𝑥 6𝑦 259) {3𝑥 6𝑦 27𝑥 2𝑦 114𝑥 5𝑦 2210) {7𝑥 3𝑦 3236

Name:Date:Unit 6 Topic 2 Homework 3: Solving Systems Using All MethodsSolve each system of equations by GRAPHING. Clearly identify your solution.1) {𝑦 4𝑥 3𝑦 3𝑥 4𝑦 2𝑥 13){𝑦 1𝑥2 952) {𝑦 2𝑥 21𝑦 2𝑥 21𝑦 3𝑥 14) {1𝑦 3𝑥 4Solve each system by SUBSTITUTION. Clearly identify your solution.𝑥 3𝑦 92𝑥 5𝑦 75) {6) {4𝑥 2𝑦 67𝑥 𝑦 837

7) {𝑥 3𝑦 245𝑥 8𝑦 58) {5𝑥 3𝑦 15𝑥 6𝑦 3Solve each system of equations by ELIMINATION. Clearly identify your solutions.𝑥 𝑦 5𝑥 5𝑦 209) {10) {𝑥 𝑦 92𝑥 7𝑦 454𝑥 3𝑦 111) {5𝑥 4𝑦 1𝑥 2𝑦 312) {2𝑥 3𝑦 538

TOPIC 3:APPLICATIONS39

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