1. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 5.1 - 1

2. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 5.1 - 2 Systems of Linear Equations Chapter 5

3. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 5.1 - 3 5.1 Systems of Linear Equations In Two Variables

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 4 5.1 Systems of Linear Equations in Two Variables Objectives 1.Solve linear systems by graphing. 2.Decide whether an ordered pair is a solution of a linear system. 3.Solve linear systems (with two equations and two variables) by substitution. 4.Solve linear systems (with two equations and two variables) by elimination. 5.Solve special systems.

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 5 5.1 Systems of Linear Equations in Two Variables Systems of Equations Suppose we wished to cut a 10-foot length of conduit into two pieces such that one piece is 4 feet longer than the other. We are looking for two numbers whose sum is 10 and whose difference is 4. If we let x represent the larger of the two numbers and y the other number, we immediately get two equations. We call such an arrangement a system of two equations in two variables.

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 6 10 5.1 Systems of Linear Equations in Two Variables Solution of a Systems of Equations A solution of a system of equations is an ordered pair that satisfies both equations at the same time. Is there such an ordered pair? Is there more then one such pair? To answer these questions, we can look at the graph of these two equations on the same coordinate system. 4 8 12 4 8 12

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 7 10 5.1 Systems of Linear Equations in Two Variables Solution of a Systems of Equations To be sure that (7, 3) is a solution of both equations, we can check by substituting 7 for x and 3 for y in both equations. 4 8 12 4 8 12 True

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 8 5.1 Systems of Linear Equations in Two Variables Solution of a Systems of Equations Given the following system of equations, determine whether the given ordered pair is a solution of the system. A solution of the system. The ordered pair must be a solution of both equations to be a solution.

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 9 5.1 Systems of Linear Equations in Two Variables Deciding Whether an ordered Pair is a Solution Given the following system of equations, determine whether the given ordered pair is a solution of the system. Not a solution of the system. Since the ordered pair is not a solution of the first equation, we need not check further; the ordered pair is not a solution.

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 10 5.1 Systems of Linear Equations in Two Variables Possible Solutions for a Linear System of Two Variables

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 11 5.1 Systems of Linear Equations in Two Variables The Substitution Method for Solving Systems

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 12 5.1 Systems of Linear Equations in Two Variables Solving a System by Substitution

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 13 5.1 Systems of Linear Equations in Two Variables Solving a System by Substitution

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 14 5.1 Systems of Linear Equations in Two Variables Solving a System by Substitution

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 15 5.1 Systems of Linear Equations in Two Variables The Elimination Method for Solving Systems

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 16 5.1 Systems of Linear Equations in Two Variables Solving a System by Elimination

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 17 5.1 Systems of Linear Equations in Two Variables Solving a System by Elimination

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 18 5.1 Systems of Linear Equations in Two Variables Solving a System by Elimination

19. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 19 5.1 Systems of Linear Equations in Two Variables Solving a System by Elimination

20. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 20 5.1 Systems of Linear Equations in Two Variables Solving a System with Fractional Coefficients

21. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 21 5.1 Systems of Linear Equations in Two Variables Solving a System with Fractional Coefficients

22. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 22 5.1 Systems of Linear Equations in Two Variables Some Special Systems Some systems of linear equations have no solution or an infinite number of solutions.

23. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 23 5.1 Systems of Linear Equations in Two Variables Some Special Systems We could get equation (1) by multiplying equation (2) by 3. Because of this, equations (1) and (2) are equivalent and have the same graph. The solution set is the set of all (infinite number of) points on the line with equation 3x + 4y = 6, written {(x, y) | 3x + 4y = 6} and read “the set of all ordered pairs (x, y) such that 3x + 4y = 6.” This is a dependent system of equations.

24. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 24 5.1 Systems of Linear Equations in Two Variables Some Special Systems The system is dependent; the lines are the same.

25. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 25 5.1 Systems of Linear Equations in Two Variables Some Special Systems

26. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 26 5.1 Systems of Linear Equations in Two Variables Some Special Systems

27. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 5.1 - 27 5.1 Systems of Linear Equations in Two Variables Some Special Systems The system is inconsistent; the lines have no points in common.