Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §3.3b 3-Var System Apps Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. MTH 55 3.3 Review § • Any QUESTIONS About • §3.3a → 3 Variable Linear Systems • Any QUESTIONS About HomeWork • §3.3a → HW-10

3. Equivalent Systems of Eqns • Operations That Produce Equivalent Systems of Equations • Interchange the position of any two eqns • Multiply (Scale) any eqn by a nonzero constant; i.e.; multiply BOTH sides • 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. 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. Gaussian Elimination • 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. • By adding appropriate multiples of the other equations, eliminate any x-terms from the second and third equations

6. Gaussian Elimination • (cont.) Rearrange the resulting two equations obtain an the Largest (in absolute value) y-term coefficient in the second equation. • 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. Gaussian Elimination • 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. • 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 • Write the solution set (Soln Triple) • Check soln in the original equations

8. Solve System by Gaussian Elim Example  Gaussian Elim • Next SCALE by using Eqn (1) as the PIVOT To Multiply • (2) by 12/6 • (3) by 12/[−5] • INTERCHANGE, or Swap, positions of Eqns (1) & (2) to get largest x-coefficient in the top equation

9. The Scaling Operation Example  Gaussian Elim • Note that the 1st 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. The Replacement Operation Yields Example  Gaussian Elim • 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 theLARGEST pivot Or

11. Our Reduced Sys Example  Gaussian Elim Or • Since | −11| > | −9.8| we do NOT need to interchange (2)↔(3) • Scale by Pivot against Eqn-(3)

12. Perform Replacement by Subtracting (3) – (2) Example  Gaussian Elim • The Hard Part is DONE • Find y & x by BACK SUBSTITUTION • From Eqn (2) • Now Easily Find the Value of z from Eqn (3)

13. BackSub into (1) Example  Gaussian Elim • x = 2 • y = −3 • z = 5 Q.E.F. • Thus the Solution Set for Our Linear System

14. Rush Hour Hours City Traffic Hours Highway Hours Total Fuel Used (gal) Week 1 2 9 3 15 Week 2 7 8 3 24 Week 3 6 18 6 34 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.

15. 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. Rush Hour Gallons City Traffic Gallons Highway Gallons Total Fuel Used (gal) Week 1 2x 9y 3z 15 Week 2 7x 8y 3z 24 Week 3 6x 18y 6z 34 Example  Fuel Usage Rates • Translate: Use Data Table • Thus theSystem of Equations

17. Example  Fuel Usage Rates • Solve by Guassian Elimination: Interchange to place largestx-Coefficient on top • Scale • Multiply Eqn (1) by −7/2 • Multiply Eqn (2) by −7/6

18. Example  Fuel Usage Rates • The new, equivalent system • Make Replacement by Adding Eqns • {Eqn (2)} + {Eqn (4)} • {Eqn (2)} + {Eqn (5)}

19. 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. Example  Fuel Usage Rates • The new, equivalent system • Now Scale Eqn (7) by the factor 47/13

21. Example  Fuel Usage Rates • The new, equivalent system • Replace by Adding: {Eqn (8)}+{Eqn (9)}

22. Example  Fuel Usage Rates • Solve Eqn (10) for z • BackSub z = 2/3 into Eqn (8) to find y

23. 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. 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. 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. 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. Example  Theater Concessions • Translate: Translate the Problem Description, Cost Eqn, and Variable Definitions into a 3 Equation System

28. 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. Example  Theater Concessions • The new, equivalent system • Make Replacement by Adding Eqns • {Eqn (1)} + {Eqn (4)} • {Eqn (1)} + {Eqn (5)}

30. Example  Theater Concessions • The new, equivalent system • pEliminated below the top Eqn • Elim d by Adding {Eqn (6)} + {Eqn (7)

31. Example  Theater Concessions • Solve Eqn (8) for c • BackSub c = 3/2 into Eqn (6) to find d

32. Example  Theater Concessions • BackSub c = 3/2 & d = 5/2 into (1) find p • The Chk is left for you to do Later

33. 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. C A B 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°.

35. Example  Missing Term • Translate: This geometric fact about triangles provides one equation: A + B + C = 180. • Translate Relationship Statements Angle B is three times the measure of angle A. B = 3A

36. Example  Missing Term • Translate Relationship Statements Angle C is 60o greater than twice the measure of A C = 60 + 2A • TranslationProduces the3-EquationSystem

37. 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. 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. 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. 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. Example  CAT Scan • Linear Attenuation Units For the Scan • Beam 1 is weakened by 0.80 units as it passes through grid cells A and B. • Beam 2 is weakened by 0.55 units as it passes through grid cells A and C. • 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. Example  CAT Scan • CAT Scan Tissue-Type Ranges LAU  Linear Attenuation Units

43. 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. Example  CAT Scan • Translate: the Attenuation Data • Beam 1 is weakened by 0.80 units as it passes through grid cells A and B. x + y = 0.80 • Beam 2 is weakened by 0.55 units as it passes through grid cells A and C x + z = 0.55 • Beam 3 is weakened by 0.65 units as it passes through grid cells B and C + z = 0.65

45. 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. Example  CAT Scan • The Replacement Operation Produces the Equivalent System • Add Equation (4) to Equation (3) to get

47. 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. Example  CAT Scan • Summarizing Results • Recall Tissue-TypeTable • 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. WhiteBoard Work • Problems From §3.3 Exercise Set • 46 • AnInconsistentSystemWHY?

50. All Done for Today CarlFriedrichGauss