Intermediate Algebra Chapter 4

1. Intermediate Algebra Chapter 4 • Systems • of • Linear Equations

2. Objective • Determine if an ordered pair is a solution for a system of equations.

3. System of Equations • Two or more equations considered simultaneously form a system of equations.

4. Checking a solution to a system of equations • 1. Replace each variable in each equation with its corresponding value. • 2. Verify that each equation is true.

5. Graphing Procedure • 1. Graph both equations in the same coordinate system. • 2. Determine the point of intersection of the two graphs. • 3. This point represents the estimated solution of the system of equations.

6. Graphing observations • Solution is an estimate • Lines appearing parallel have to be checked algebraically. • Lines appearing to be the same have to be checked algebraically.

7. Classifying Systems • Meet in Point – Consistent – independent • Parallel – Inconsistent – Independent • Same – Consistent - Dependent

8. Def: Dependent Equations • Equations with identical graphs

9. Independent Equations • Equations with different graphs.

10. Algebraic Check • Same Line

11. Algebraic Check • Parallel Lines

12. Algebraic Check • Meet in a point {(x,y)}

13. Calculator Method for Systems • Solve each equation for y • Input each equation into Y= • Graph • Set Window • Use CalIntersect

14. Calculator Problem

15. Calculator Problem 2

16. Calculator Problem 3

17. Objective • Solve a System of Equations using the Substitution Method.

18. Substitution Method • 1. Solve one equation for one variable • 2. In other equation, substitute the expression found in step 1 for that variable. • 3. Solve this new equation (1 variable) • 4. Substitute solution in either original equation • 5. Check solution in original equation.

19. Althea Gibson – tennis player • “No matter what accomplishments you make, someone helped you.”

20. Intermediate Algebra • The • Elimination • Method

21. Notes on elimination method • Sometimes called addition method • Goal is to eliminate on of the variables in a system of equations by adding the two equations, with the result being a linear equation in one variable.

22. 1. Write both equations in ax + by = c form • 2. If necessary, multiply one or both of the equations by appropriate numbers so that the coefficients of one of the variables are opposites.

23. Procedure for addition method cont. • 3. Add the equations to eliminate a variable. • 4. Solve the resulting equation • 5. Substitute that value in either of the original equations and solve for the other variable. • 6. Check the solution.

24. Procedure for addition method cont. • Solution could be ordered pair. • If a false statement results i.e. 1 = 0, then lines are parallel and solution set is emptyset. (inconsistent) • If a true statement results i.e. 0 = 0, then lines are same and solution set is the line itself. (dependent)

25. Practice Problem • Answer {(2,4)}

26. Practice Problem Hint: eliminate x first • Answer {(-7/2,-4)}

27. Practice Problem

28. Practice problem

29. Special Note on Addition Method • Having solved for one variable, one can eliminate the other variable rather than substitute. • Useful with fractions as answers.

30. Practice Problem – eliminate one variable and than the other • Answer: {(8/3,1/3)}

31. Confucius • “It is better to light one small candle than to curse the darkness.”

32. Intermediate Algebra 4.2 • Systems • Of • Equations • In • Three Variables

33. Objective • To use algebraic methods to solve linear equations in three variables.

34. Def: linear equation in 3 variables • is any equation that can be written in the standard form ax + by +cz =d where a,b,c,d are real numbers and a,b,c are not all zero.

35. Def: Solution of linear equation in three variables • is an ordered triple (x,y,z) of numbers that satisfies the equation.

36. Procedure for 3 equations, 3 unknowns • 1. Write each equation in the form ax +by +cz=d • Check each equation is written correctly. • Write so each term is in line with a corresponding term • Number each equation

37. Procedure continued: • 2. Eliminate one variable from one pair of equations using the elimination method. • 3. Eliminate the same variable from another pair of equations. • Number these equations

38. Procedure continued • 4. Use the two new equations to eliminate a variable and solve the system. • 5. Obtain third variable by back substitution in one of original equations

39. Procedure continued • Check the ordered triple in all three of the original equations.

40. Sample problem 3 equations

41. Answer to 3 eqs-3unknowns • {(-2,3,1)}

42. Bertrand Russell – mathematician (1872-1970) • “Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform.”

43. Cramer’s Rule • Objective: Evaluate determinants of 2 x 2 matrices • Objective: Solve systems of equations using Cramer’s Rule

44. Determinant

45. Cramer’s rule intuitive • Each denominator, D is the determinant of a matrix containing only the coefficients in the system. To find D with respect to x, we replace the column of s-coefficients in the coefficient matrix with the constants form the system. To find D with respect to y, replace the column of y-coefficients in the coefficient matrix sit the constant terms.

46. Sample Problem: Evaluate: • Answer = 16

47. Sample Cramer’s Rule problem • Solve by Cramer’s Rule

48. Cramer’s Rule Answer

49. Senecca • “It is not because things are difficult that we do not dare, it is because we do not dare that they are difficult.”

50. Intermediate Algebra 5.5 • Applications • Objective: Solve application problems using 2 x 2 and 3 x 3 systems.