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Solving Systems of Equations Graphically, Substitution, and Elimination Methods

Learn how to solve systems of linear equations by graphing, substitution, and elimination methods. Understand consistency, inconsistency, and dependence of systems. Practice with exercises to master these techniques.

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Solving Systems of Equations Graphically, Substitution, and Elimination Methods

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  1. Section 8.1 Solving Systems of Linear Equations by Graphing

  2. Solve a system of two equations in two variables by graphing. A OBJECTIVES

  3. B OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent.

  4. C OBJECTIVES Solve an application.

  5. SOLVING SYSTEMS OF SIMULTANEOUS EQUATIONS There are three possible solutions.

  6. POSSIBLE SOLUTION 1 Consistent and independent equations: The graphs of the equations intersect at one point, whose coordinates give the solution of the system.

  7. POSSIBLE SOLUTION 2 Inconsistent equations: The graphs of the equations are parallel lines; there is no solution for the system

  8. POSSIBLE SOLUTION 3 Dependent equations: The graphs of the equations coincide (are the same). There are infinitely many solutions for the system.

  9. COMPARISONS Intersecting Lines Have different slopes

  10. COMPARISONS Intersecting Lines Have one solution

  11. COMPARISONS Intersecting Lines Form a consistent system

  12. COMPARISONS Parallel Lines Have the same slopes

  13. COMPARISONS Parallel Lines Have different y-intercepts

  14. COMPARISONS Parallel Lines Have no solution

  15. COMPARISONS Parallel Lines Form inconsistent systems

  16. COMPARISONS Coinciding Lines Have the same slope

  17. COMPARISONS Coinciding Lines Have the samey-intercept

  18. COMPARISONS Coinciding Lines Have infinite solutions

  19. COMPARISONS Coinciding Lines Form a dependent system

  20. A HELPFUL HINT For

  21. A HELPFUL HINT For

  22. Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.1

  23. Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.1Exercise #1

  24. Use the graphical method to solve the system.

  25. x y0 24 0 Use the graphical method to solve the system.

  26. x y0 04 2 Use the graphical method to solve the system.

  27. Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.1Exercise #2

  28. Use the graphical method to solve the system.

  29. x y0 2–1 0 Use the graphical method to solve the system.

  30. x y0 4–2 0 Use the graphical method to solve the system. Inconsistent:No Solution.

  31. Section 8.2 Solving Systems of Equations by Substitution

  32. Solve a system of equations in two variables. A OBJECTIVES

  33. B OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent.

  34. C OBJECTIVES Solve an application.

  35. PROCEDURE: Solving a system of equations by the Substitution Method. 1. Solve one of the equations for x or y.

  36. PROCEDURE: Solving a system of equations by the Substitution Method. Substitute the resulting expression into the other equation.

  37. PROCEDURE: Solving a system of equations by the Substitution Method. 3. Solve the new equation for the variable.

  38. PROCEDURE: Solving a system of equations by the Substitution Method. 4. Substitute the value of the variable and solve to get the value for the second variable.

  39. PROCEDURE: Solving a system of equations by the Substitution Method. 5. Check the solution by substituting the numerical values of the variables in both equations.

  40. Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.2Exercise #3

  41. Use the method of substitution to solve the system (if possible). No Solution (inconsistent)

  42. Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.2Exercise #4

  43. Use the method of substitution to solve the system (if possible). Dependent (infinitely many solutions).

  44. Section 8.3 Solving Systems of Equations by Elimination

  45. Solve a system of equations in two variables. A OBJECTIVES

  46. B OBJECTIVES Determine whether a system of equations is consistent, inconsistent, or dependent.

  47. C OBJECTIVES Solve an application.

  48. ELIMINATION METHOD One or both equations in a system of simultaneous equations can be multiplied (or divided) by any nonzero number to obtain an equivalent system.

  49. ELIMINATION METHOD In the equivalent system, the coefficients of x (or y) are opposites, thus eliminatingx or y when the equations are added.

  50. Chapter 8 Solving Systems of Linear Equations and Inequalities Section 8.3Exercise #5

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