Mathematics 116 Chapter 5 Bittinger

1. Mathematics 116 Chapter 5Bittinger • Linear Systems • and • Matrices

2. Mark Twain - American author (1835-1910) • “What is the most rigorous law of our being? Growth. No smallest atom of our moral, mental, or physical structure can stand still a year. It grows, it must grow…nothing can prevent it.”

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

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

5. 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.

6. Solving systems of equations • Solve Graphically • Graph both equations with appropriate window. • Determine point of intersection using intersect feature of calculator • Graph by hand – more points, more care, the more accurate

7. Graphical Analysis3 Possibilities • Two lines meet in a point • Two lines are parallel • Two lines are the same

8. Meet in a Point • Consistent – has a single solution at the point of intersection • Independent – graphs are different and intersect at one point. • They have different slopes. • Solution is an ordered pair

9. Parallel Lines • Inconsistent system – the system has no solution. • Solution set • Independent – the graphs are different • Lines are parallel • Slopes are the same • y intercept is different

10. Same Line • Consistent System – the system has an infinite number of solutions • Dependent – the graphs are identical • Have the same slope • Have the same y intercept

11. 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.

12. 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.

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

14. Def: Dependent Equations • Equations with identical graphs

15. Def: Dependent Equations • Equations with identical graphs

16. Calculator Problem

17. Calculator Problem 2

18. Calculator Problem 3

19. Algebraic Analysis

20. Same line

21. Same line example • 2x + 3y = 4 • 4x + 6y = 8 • Solution set

22. Parallel Lines

23. Parallel lines example • 2x + 3y = 4 • 4x + 6y = 5

24. Solve using substitution • Isolate one of the variables • In the other equation, substitute the expression • Solve the new equation for one unknown • Substitute the value obtained and solve for the other variable • Check the result

25. Substitution: Special Notes For Lines • solution is ordered pair - Two lines • Obtain false statement such as 0 = 1 • ----parallel lines – solution set is empty set • Obtain true statement – such as 0 = 0 • This is the same line and the solution set is the line itself.

26. Objective • Use systems of equations to model and solve real-life problems.

27. Break Even • C = Total Cost = cost per unit *number of units +initial cost • R = Total Revenue = Price per unit * # of units • Break Even is R = C

28. Theodore Roosevelt • “I think we consider too much the good luck of the early bird and not enough the bad luck of the early worm.”

29. Mathematics 116 • Systems of Linear Equations • In • Two Variables

30. Solve by Elimination • Write in standard form • Clear equations of fractions or decimals • Multiply one or both equation by number(s) so that a pair of terms are additive inverses • Add the equations • Solve for one unknown • Substitute to find other unknown • Check

31. Elimination: Special Notes • solution is ordered pair - Two lines • Obtain false statement such as 0 = 1 • ----parallel lines – solution set is empty set • Obtain true statement – such as 0 = 0 • This is the same line and the solution set is the line itself

32. Practice Problem • Answer {(2,4)}

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

34. Dale Earnhardt • “You win some, you lose some, you wreck some.”

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

36. College Algebra • Systems • Of • Equations • In • Three Variables

37. 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.

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

39. 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

40. 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

41. 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

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

43. Sample problem 3 equations

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

45. 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.”

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

47. Mixture Problems • ****Use table or chart • Include all units • Look back to test reasonableness of answer.

48. Sample Problem • How many milliliters of a 10% HCl solution and 30% HCl solution must be mixed together to make 200 milliliters of 15% HCl solution?

49. Mixture problem equations

50. Mixture problem answers • 150 mill of 10% sol • 50 mill of 30% sol • Gives 200 mill of 15% sol