1. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §3.3b 3-Var System Apps

2. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §3.3a → 3 Variable Linear Systems  Any QUESTIONS About HomeWork §3.3a → HW-10 3.3 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 3 Bruce Mayer, PE Chabot College Mathematics Equivalent Systems of Eqns  Operations That Produce Equivalent Systems of Equations 1.Interchange the position of any two eqns 2.Multiply (Scale) any eqn by a nonzero constant; i.e.; multiply BOTH sides 3.Add a nonzero multiple of one eqn to another to affect a Replacment  A special type of Elimination called Gaussian Elimination uses these steps to solve multivariable systems

4. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 4 Bruce Mayer, PE Chabot College Mathematics Gaussian Elimination  An algebraic method used to solve systems in three (or more) variables.  The original system is transformed to an equivalent one of the form: Ax + By + Cz = D Ey + Fz = G Hz = K  The third eqn is solved for z and back- substitution is used to find y and then x

5. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 5 Bruce Mayer, PE Chabot College Mathematics Gaussian Elimination 1.Rearrange, or InterChange, the equations, if necessary, to obtain the Largest (in absolute value) x-term coefficient in the first equation. The Coefficient of this large x-term is called the leading-coefficient or pivot-value. 2.By adding appropriate multiples of the other equations, eliminate any x-terms from the second and third equations

6. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 6 Bruce Mayer, PE Chabot College Mathematics Gaussian Elimination 2.(cont.) Rearrange the resulting two equations obtain an the Largest (in absolute value) y-term coefficient in the second equation. 3.If necessary by adding appropriate multiple of the third equation from Step 2, eliminate any y-term from the third equation. Solve the resulting equation for z.

7. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 7 Bruce Mayer, PE Chabot College Mathematics Gaussian Elimination 4.Back-substitute the values of z from Steps 3 into one of the equations in Step 3 that contain only y and z, and solve for y. 5.Back-substitute the values of y and z from Steps 3 and 4 in any equation containing x, y, and z, and solve for x 6.Write the solution set (Soln Triple) 7.Check soln in the original equations

8. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Gaussian Elim  Solve System by Gaussian Elim  INTERCHANGE, or Swap, positions of Eqns (1) & (2) to get largest x-coefficient in the top equation  Next SCALE by using Eqn (1) as the PIVOT To Multiply (2) by 12/6 (3) by 12/[−5]

9. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Gaussian Elim  The Scaling Operation  Note that the 1 st Coeffiecient in the Pivot Eqn is Called the Pivot Value The Pivot is used to SCALE the Eqns Below it  Next Apply REPLACEMENT by Subtracting Eqs (2) – (1) (3) – (1)

10. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Gaussian Elim  The Replacement Operation Yields Or  Note that the x-variable has been ELIMINATED below the Pivot Row Next Eliminate in the “y” Column  We can use for the y-Pivot either of −11 or −9.8 For the best numerical accuracy choose the LARGEST pivot

11. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Gaussian Elim  Our Reduced Sys  Since | −11| > | −9.8| we do NOT need to interchange (2)↔(3)  Scale by Pivot against Eqn-(3) Or

12. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Gaussian Elim  Perform Replacement by Subtracting (3) – (2)  Now Easily Find the Value of z from Eqn (3)  The Hard Part is DONE  Find y & x by BACK SUBSTITUTION  From Eqn (2)

13. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Gaussian Elim  BackSub into (1)  Thus the Solution Set for Our Linear System  x = 2  y = −3  z = 5

14. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Useage Rates  A food service distributor conducted a study to predict fuel usage for new delivery routes, for a particular truck. Use the chart to find the rates of fuel consumption in rush hour traffic, city traffic, and on the highway. 6 3 3 Highway Hours 34186Week 3 2487Week 2 1592Week 1 Total Fuel Used (gal) City Traffic Hours Rush Hour Hours

15. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  Familiarize: The Fuel Use Calc’d by the RATE Eqn: Quantity = (Rate)·(Time) = (Time)·(Rate)  In this Case the Rate Eqn (UseTime)·(UseRate) → (hr)·(Gal/hr) So LET: –x ≡ Fuel Use Rate (Gal/hr) in Rush Hr Traffic –y ≡ Fuel Use Rate (Gal/hr) in City Traffic –z ≡ Fuel Use Rate (Gal/hr) in HiWay Traffic

16. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  Translate: Use Data Table 6z 3z3z 3z3z Highway Gallons 3418y6x6xWeek 3 248y8y7x7xWeek 2 159y9y2x2xWeek 1 Total Fuel Used (gal) City Traffic Gallons Rush Hour Gallons  Thus the System of Equations

17. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  Solve by Guassian Elimination: Interchange to place largest x-Coefficient on top  Scale Multiply Eqn (1) by −7/2 Multiply Eqn (2) by −7/6

18. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  The new, equivalent system  Make Replacement by Adding Eqns {Eqn (2)} + {Eqn (4)} {Eqn (2)} + {Eqn (5)}

19. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  The new, equivalent system  Notice how x has been Eliminated below the top Eqn  Clear Fractions by multiplying Eqn (6) by −2

20. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  The new, equivalent system  Now Scale Eqn (7) by the factor 47/13

21. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  The new, equivalent system  Replace by Adding: {Eqn (8)}+{Eqn (9)}

22. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  Solve Eqn (10) for z  BackSub z = 2/3 into Eqn (8) to find y

23. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  BackSub z = 2/3 and y = 1 into Eqn (2) to find x  Chk x = 2, y = 1 & z = 2/3 in Original Eqns

24. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Fuel Usage Rates  Continue Chk of x = 2, y = 1 & z = 2/3  State: The Delivery Truck Uses 2 Gallons per Hour in Rush Hour traffic 1 Gallons per Hour in City traffic 2/3 Gallons per Hour in HighWay traffic 

25. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  At a movie theatre, Kara buys one popcorn, two drinks and 2 candy bars, all for $12. Jaypearl buys two popcorns, three drinks, and one candy bar for $17. Nyusha buys one popcorn, one drink and three candy bars for $11. Find the individual cost of one popcorn, one drink and one candy bar

26. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  Familiarize: Allow UNITS to guide us to the Total Cost Equation:  This Eqn does yield the Total Cost as required. Thus LET c ≡ The UnitCost of Candy Bars d ≡ The UnitCost of Soft Drinks p ≡ The UnitCost of PopCorn Buckets

27. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  Translate: Translate the Problem Description, Cost Eqn, and Variable Definitions into a 3 Equation System

28. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  Solve by Guassian Elim: Interchange to place largest x-Coefficient on top  Scale Multiply Eqn (2) by −2 Multiply Eqn (3) by −2

29. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  The new, equivalent system  Make Replacement by Adding Eqns {Eqn (1)} + {Eqn (4)} {Eqn (1)} + {Eqn (5)}

30. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  The new, equivalent system  p Eliminated below the top Eqn  Elim d by Adding {Eqn (6)} + {Eqn (7)

31. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  Solve Eqn (8) for c  BackSub c = 3/2 into Eqn (6) to find d

32. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 32 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  BackSub c = 3/2 & d = 5/2 into (1) find p  The Chk is left for you to do Later

33. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 33 Bruce Mayer, PE Chabot College Mathematics Example  Theater Concessions  A Quick Summary  State: The Cost for the Movie Theater Concessions: $4.00 for a Tub of PopCorn $2.50 for a Soft Drink $1.50 for a Candy Bar

34. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 34 Bruce Mayer, PE Chabot College Mathematics Example  Missing Term  In triangle ABC, the measure of angle B is three times the measure of angle A. The measure of angle C is 60° greater than twice the measure of angle A. Find the measure of each angle.  Familiarize: Make a sketch and label the angles A, B, and C. Recall that the measures of the angles in any triangle add to 180°. A B C

35. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  Missing Term  Translate: This geometric fact about triangles provides one equation: A + B + C = 180. B = 3A Angle B is three times the measure of angle A.  Translate Relationship Statements

36. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 36 Bruce Mayer, PE Chabot College Mathematics Example  Missing Term  Translate Relationship Statements C = 60 + 2A Angle C is 60 o greater than twice the measure of A  Translation Produces the 3-Equation System

37. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  Missing Term  Since this System has Missing Terms in two of the Equations, Substitution is faster than Elimination  Sub into Top Eqn B = 3A C = 60+2A

38. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 38 Bruce Mayer, PE Chabot College Mathematics Example  Missing Term  BackSub A = 20° into the other eqns  Check → 20° + 60° + 100° = 180°   State: The angles in the triangle measure 20°, 60°, and 100°

39. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 39 Bruce Mayer, PE Chabot College Mathematics Example  Missing Term  In triangle ABC, the measure of angle B is three times the measure of angle A. The measure of angle C is 60° greater than twice the measure of angle A.

40. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 40 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  Let A, B, and C be three grid cells as shown  A CAT scanner reports the data on the following slide for a patient named Satveer

41. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 41 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  Linear Attenuation Units For the Scan i.Beam 1 is weakened by 0.80 units as it passes through grid cells A and B. ii.Beam 2 is weakened by 0.55 units as it passes through grid cells A and C. iii.Beam 3 is weakened by 0.65 units as it passes through grid cells B and C  Using the following table, determine which grid cells contain each of the type of tissue listed

42. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 42 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  CAT Scan Tissue-Type Ranges LAU  Linear Attenuation Units

43. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 43 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  Familiarize: Suppose grid cell A weakens the beam by x units, grid cell B weakens the beam by y units, and grid cell C weakens the beam by z units.  Thus LET: x ≡ The Cell-A Attenuation y ≡ The Cell-B Attenuation z ≡ The Cell-C Attenuation

44. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 44 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  Translate: the Attenuation Data i.Beam 1 is weakened by 0.80 units as it passes through grid cells A and B. x + y = 0.80 ii.Beam 2 is weakened by 0.55 units as it passes through grid cells A and C x + z = 0.55 iii.Beam 3 is weakened by 0.65 units as it passes through grid cells B and C + z = 0.65

45. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 45 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  Thus the Equation System  Even with Missing Terms Elimination is sometimes a good solution method  Add −1 times Equation (1) to Equation (2)

46. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 46 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  The Replacement Operation Produces the Equivalent System  Add Equation (4) to Equation (3) to get

47. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 47 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  Back-substitute z = 0.20 into Eqn (4) to Obtain  Back-substitute y = 0.45 into Eqn (1) and solve for x

48. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 48 Bruce Mayer, PE Chabot College Mathematics Example  CAT Scan  Summarizing Results  Recall Tissue-Type Table  Thus Conclude Cell A contains tumorous tissue (x = 0.35) Cell B contains a bone (y = 0.45) Cell C contains healthy tissue (z = 0.20)

49. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 49 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §3.3 Exercise Set 46  An Inconsistent System WHY?

50. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 50 Bruce Mayer, PE Chabot College Mathematics All Done for Today Carl Friedrich Gauss

51. BMayer@ChabotCollege.edu MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 51 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –