Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:

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1 Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system: Example 1: Use the graph to solve the system. Then check your solution algebraically. 2x + y = 4 Equation 1 3x 5y = 6 Equation 2

2 Example 2: Solve the linear systems by graphing. y = 3x 2 Equation 1 y = -2x + 3 Equation 2 x 3y = 2 Equation 1 5x + y = 4 Equation 2

3 Name Date LESSON 6.1 Practice A For use with the lesson Solve Linear Systems by Graphing Tell whether the ordered pair is a solution of the linear system. 1. (0, 4); x + y = 4 x 5y = (3, 3); x + 2y = 9 4x y = (1, 2); 2x 3y = 8 3x + 2y = 1 Solve the linear system by graphing. Check your solution. 4. y = x y = -2x + 1 y = x 2 y = x - 2

4 Lesson 6.2 Solve Linear Systems by Substitution Example 1: Solve the linear system using substitution. y = 2x 3 Equation 1 x + 3y = 5 Equation 2 Example 2: Solve the linear system using substitution. 2x + 5y = 5 Equation 1 x 4y = 9 Equation 2

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6 Name Date LESSON 6.2 Practice A For use with the lesson Solve Linear Systems by Substitution Solve for the indicated variable. 1. 9x + y = 7; y 2. 3x y = 10; y 3. x 4y = 1; x Solve the linear system by using substitution. 4. x = 1 y y = 2x 2 5. x = 4y + 14 y = 3x + 3

7 6. y = 3x 1 4x + 3y = 2 7. y = 2x + 4 5y 2x = x 2y = 14 x = 10 6y

8 opyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. Name CHAPTER 6 Practice Quiz 1 For use after the lessons Solve Linear Systems by Graphing and Solve Linear Systems by Substitution Solve the linear system by graphing. 1. x y = 2 2. x + 2y = 6 x + y = 4 3x + y = x + 3y = 9 x + 2y = 6

9 Solve the linear system using substitution. 4. 4x 2y = 14 x = 10 6y 5. x = 1 y y = 2x 2 6. x + 2y = 6 7x + 3y = 8

10 Solve the linear system using substitution. 7. x + 2y = 11 3x 4y = x + y = 8 x + 2y = 5

11 Lesson 6.3 Solve Linear Systems by Adding or Subtracting Example 1: Solve the linear system by adding or subtracting. 2x + 4y = 2 Equation 1 4x 4y = 16 Equation 2 Example 2: Solve the linear system by adding or subtracting. 7x + 5y = 18 Equation 1 7x 3y = 34 Equation 2 Example 3: Solve the linear system by adding or subtracting.

12 6x 4y = 10 Equation 1 13y = 6x + 8 Equation 2

13 Name Date LESSON 6.3 Practice A For use with the lesson Solve Linear Systems by Adding or Subtracting Rewrite the linear system so that the like terms are arranged in columns. 1. 3x y =23 y + 8x = x = y + 1 3y + 8x = x 4y = 8 4y = 7x + 9 Solve the linear system by using elimination. 4. 6x y = 5 3x + y = 4 5. x + 4y = 9 x 2y = 3

14 6. 5x 3y = 14 x + 3y = x + y = 7 x + y = x + 3y = 18 4x 2y = x 3y = 36 5x = 3y + 30

15 Lesson 6.4: Solve Linear Systems by Multiplying First Least Common Multiple: Example 1: Solve the linear system. 3x 2y = 4 Equation 1 7x 4y = 6 Equation 2 Example 2: Solve the linear system. 5x +2y = 18 Equation 1 7y = 3x + 19 Equation 2

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17 Name Date LESSON 6.4 Practice A For use with the lesson Solve Linear Systems by Multiplying First Describe the first step you would use to solve the linear system. 1. x + y = 4 3x 7y = x + 6y = 1 4x + 7y = x 6y = 1 x + y = 4 Solve the linear system by using elimination. 4. x + y = 3 2x + 4y = x + y = 8 3x + 3y = x y = 10 2x + 5y = x 4y = 42 x 6y = 24

18 Name CHAPTER 6 Practice Quiz 2 For use after the lessons Solve Linear Systems by Adding or Subtracting and Solve Linear Systems by Multiplying First Solve the linear system using elimination. 1. 7x + y = 6 6x y = 7 Answers x + 5y = 1 2x + y = x + 4y = 10 x + 2y = x + 2y = 2 6x + 3y = 5

19 Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 5. 5x y = 7 4x y = 6 Answers x + y = 3 2x + 4y = x + y = 8 3x + 3y = x 4y = 42 x 6y = 24

20 Lesson 6.5: Solve Special Types of Linear Systems Inconsistent system: Consistent dependent system: Example 1: Show that the linear system has no solution. 5x + 4y = 16 Equation 1 5x 4y = 8 Equation 2 Example 2: Show that the linear system has infinitely many solutions. 2 y = x +5 Equation 1 3 2x + 3y = 15 Equation 2

21 Example 3: Without solving the linear system, tell whether the linear system has one solution, no solution, or infinitely many solutions. 7x 2y = 9 Equation 1 3x + y = 10 Equation 1 7x 2y = 1 Equation 2 6x 2y = 20 Equation 2 Number of Solutions? What are the lines doing? Characteristics of the lines

22 Name Date LESSON 6.5 Practice A For use with the lesson Solve Special Types of Linear Systems Identify the slope of the linear equation. 1. y 3x = x + 2y = x 3y = 15 Graph the linear system. Then use the graph to tell whether the linear system has one solution, no solution, or infinitely many solutions. 4. x + y = 4. y = x y 2x = 3 x + y = 2

23 6. 2x +2y = 4 y = x x y = 1 2x + y = x + 2y = 8 3x y = 3

24 Lesson 6.6 Solve Systems of Linear Inequalities System of linear inequalities: Solution of a system of linear inequalities: Graph of a system of linear inequalities: Example 1: Graph the system of inequalities. 1 y < x + 2 Inequality 1 2 y 2x + 5 Inequality 2 Example 2: Graph the system of inequalities. y 5 Inequality 1 x < 4 Inequality 2 y 2x + 2 Inequality 3

25 Example 3: Write a system of inequalities for the shaded region.

26 Name Date LESSON 6.6 Practice A For use with the lesson Solve Systems of Linear Inequalities Tell whether the ordered pair is a solution of the system of inequalities. 1. (2, 1) 2. ( 3, 2) Graph the system of inequalities. 3. x + y 4 4. x + y 4 x < 2 x < 2

27 5. x y y + x 4 y > 2 y < 2 7. x y 4 8. y + x 4 x > 2 y < 2

28 Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. Name CHAPTER 6 Practice Quiz 3 For use after the lessons Solve Special Types of Linear Systems and Solve Systems of Linear Inequalities Graph the linear system. Then use the graph to tell whether the linear system has one solution, no solution, or infinitely many solutions. 1. 2y 5x = x + 2y = 8 2y x = 4 3x y = 5 Answers x + y = 4 4x 2y = 8 Graph the system of inequalities. 4. x > 4 5. y x + 3 y < 2 y x 1 7. y x y > 2x y > 2 x > 3

29 Name Date CHAPTER 6 Practice Test For use after the chapter Systems of Equations and Inequalities Graph to solve the linear system. 1. y = (-1/2)x y = (2/3)x + 2 y = 2x + 1 x = 3 Answers Solve the linear system using substitution. 3. x = 4 4. y = 2x x + 2y = 4 x = - 2y 5. 3x + 2 = y 6. y = 2x 3 x = 11 2y x + 3y = 5

30 reserved. Name Date CHAPTER 6 Practice Test continued For use after the chapter Systems of Equations and Inequalities Solve the linear system using elimination. 7. x + y = x + 4y = -9 x y = 12 3x + y = 0 Answers x 4y = x + 2y = 5 4x + 2y = 6 4x 3y = 16 Determine whether the linear system has one solution, no solution, or infinitely many solutions. 11. y = 5x y = - 2x x y = 10 y = 5x + 1 y = 2x + 8 y = 6x 10 Graph the system of linear inequalities. 14. y > y > - 1 x < 3

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