Chabot Mathematics §3.3b 3-Var System Apps Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot College Mathematics 1 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Review § 3.3 MTH 55 Any QUESTIONS About • §3.3a → 3 Variable Linear Systems Any QUESTIONS About HomeWork • §3.3a → HW-10 Chabot College Mathematics 2 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 3 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 backsubstitution is used to find y and then x Chabot College Mathematics 4 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 5 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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. Chabot College Mathematics 6 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 7 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Gaussian Elim Solve System by Gaussian Elim 1 2 3 6 x 3 y 4 z 41 12 x 5 y 7 z 26 5 x 2 y 6 z 14 INTERCHANGE, or Swap, positions of Eqns (1) & (2) to get largest x-coefficient in the top equation Chabot College Mathematics 8 1 2 3 12 x 5 y 7 z 26 6 x 3 y 4 z 41 5 x 2 y 6 z 14 Next SCALE by using Eqn (1) as the PIVOT To Multiply • (2) by 12/6 • (3) by 12/[−5] Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Gaussian Elim The Scaling Operation 1 12 x 5 y 7 z 26 12 2 6 x 3 y 4 z 41 6 12 5 x 2 y 6 z 14 3 5 1 2 3 12 x 5 y 7 z 26 12 x 6 y 8 z 82 12 x 4.8 y 14.4 z 33.6 Chabot College Mathematics 9 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) Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Gaussian Elim The Replacement Operation Yields 1 2 3 12 x 5 y 7 z 26 0 x 11y 15 z 108 0 x 9.8 y 7.4 z 7.6 Or 1 2 3 12 x 5 y 7 z 11y 15 z 108 9.8 y 7.4 z 7.6 Chabot College Mathematics 10 26 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 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Gaussian Elim Our Reduced Sys 1 2 3 11y 15 z 108 1 2 9.8 y 7.4 z 7.6 3 12 x 5 y 7 z 26 12 x 5 y 7 z 26 11y 15 z 108 11 9.8 y 7.4 z 7.6 9.8 Or Since | −11| > | −9.8| we do NOT need to 1 12 x 5 y 7 z interchange (2)↔(3) Scale by Pivot against Eqn-(3) Chabot College Mathematics 11 2 3 11y 15 z 26 108 11y 8.306 z 8.531 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Gaussian Elim Perform Replacement by Subtracting (3) – (2) 1 2 3 12 x 5 y 7 z 11y 15 z 26 Find y & x by BACK SUBSTITUTION 108 From Eqn (2) 23.306 z 116.531 Now Easily Find the Value of z from Eqn (3) z 116.531 23.306 5 Chabot College Mathematics 12 The Hard Part is DONE 108 15 z 108 75 y 11 11 y 33 11 3 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Gaussian Elim BackSub into (1) 1 2 3 6 x 3 y 4 z 41 12 x 5 y 7 z 26 12 x 5 y 7 z 26 7 z 5 y 26 x 5 x 2 y 6 z 14 12 35 15 26 24 x 2 12 12 x=2 Thus the Solution Set for Our Linear System Chabot College Mathematics 13 y = −3 z=5 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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. 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 Chabot College Mathematics 14 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 15 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates Translate: Use Data Table Rush Hour Gallons Week 1 2x Week 2 7x Week 3 6x City Traffic Gallons 9y 8y 18y Highway Gallons 3z 3z 6z Total Fuel Used (gal) 15 24 34 2 x 9 y 3z 15 1 Thus the System of 7 x 8 y 3z 24 2 Equations 6 x 18 y 6 z 34 3 Chabot College Mathematics 16 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates Solve by Guassian Elimination: Interchange to place largest x-Coefficient on top 7x 8y 3z 24 2x 9y 3z 15 6 x 18 y 6 z 34 2 1 3 Scale • Multiply Eqn (1) by −7/2 • Multiply Eqn (2) by −7/6 Chabot College Mathematics 17 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates The new, equivalent system 8y 3z 24 63 21 105 7x y z 2 2 2 119 7 x 21 y 7 z 3 7x 2 4 5 Make Replacement by Adding Eqns • {Eqn (2)} + {Eqn (4)} • {Eqn (2)} + {Eqn (5)} Chabot College Mathematics 18 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates The new, equivalent system 7x 8y 3z 24 47 15 57 y z 2 2 2 47 13 y 4 z 3 2 6 7 Notice how x has been Eliminated below the top Eqn Clear Fractions by multiplying Eqn (6) by −2 Chabot College Mathematics 19 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates The new, equivalent system 7x 8y 47 y 3z 15 z 24 57 47 13 y 4 z 3 2 8 7 Now Scale Eqn (7) by the factor 47/13 47 47 188 2209 13 y 4 z 47 y z 13 3 13 39 Chabot College Mathematics 20 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates The new, equivalent system 7x 2 8 9 3z 24 15 z 57 188 2209 47 y z 13 39 8y 47 y Replace by Adding: {Eqn (8)}+{Eqn (9)} 7 x 8 y 3z 15 47 y 15 z 57 7 14 z 13 39 Chabot College Mathematics 21 2 8 10 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates Solve Eqn (10) for z 14 14 13 2 1 2 13 7 z z z 39 39 7 3 1 3 7 13 BackSub z = 2/3 into Eqn (8) to find y 2 47 y 15 57 47 y 10 57 3 47 47 y 57 10 47 y y 1 47 Chabot College Mathematics 22 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates BackSub z = 2/3 and y = 1 into Eqn (2) to find x 7 x 81 3 2 24 7 x 8 2 24 3 7 x 24 10 7 x 14 x 2 Chk x = 2, y = 1 & z = 2/3 in Original Eqns 72 81 22 91 62 181 Chabot College Mathematics 23 2 3 24 3 2 3 15 3 2 6 34 3 2 1 3 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Fuel Usage Rates Continue Chk of x = 2, y = 1 & z = 2/3 14 8 2 24 24 24 9 2 15 15 15 4 12 18 4 34 34 34 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 Chabot College Mathematics 24 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 25 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Theater Concessions Familiarize: Allow UNITS to guide us to the Total Cost Equation: $ $ $ No. CBars $Cost No. Drinks No. PopCorn Drink CBar PopCorn 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 Chabot College Mathematics 26 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Theater Concessions Translate: Translate the Problem Description, Cost Eqn, and Variable Definitions into a 3 Equation System $ $ $ No. Drinks No. PopCorn No. CBars $Cost Drink CBar PopCorn p 2d 2 p 3d p Chabot College Mathematics 27 d 2c $12 Kara $17 Jaypearl 3c $11 Nyusha c Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Theater Concessions Solve by Guassian Elim: Interchange to place largest x-Coefficient on top 2 p 3d c 17 p 2d 2c 12 p 3c 11 d 1 2 3 Scale • Multiply Eqn (2) by −2 • Multiply Eqn (3) by −2 Chabot College Mathematics 28 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Theater Concessions The new, equivalent system 2p 3d c 17 2 p 4d 4c 24 2 p 2d 6c 22 1 4 5 Make Replacement by Adding Eqns • {Eqn (1)} + {Eqn (4)} • {Eqn (1)} + {Eqn (5)} Chabot College Mathematics 29 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Theater Concessions The new, equivalent system 2p 3d 17 c d 3c 7 5c 5 d 1 6 7 p Eliminated below the top Eqn Elim d by Adding {Eqn (6)} + {Eqn (7) 2p 17 d 3c 7 8c 12 Chabot College Mathematics 30 3d c 1 6 8 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Theater Concessions Solve Eqn (8) for c 8c 12 8 3 c c $1.50 2 BackSub c = 3/2 into Eqn (6) to find d 3 14 d 3c 7 d 3 2 2 14 9 5 d d $2.50 2 2 2 Chabot College Mathematics 31 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Theater Concessions BackSub c = 3/2 & d = 5/2 into (1) find p 15 3 5 3 2 p 3 17 2 p 17 2 2 2 2 15 3 34 34 18 2p 2p 2 2 2 2 2 16 1 2p 8 2 p 8 2 2 8 p p $4.00 2 The Chk is left for you to do Later Chabot College Mathematics 32 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Theater Concessions A Quick Summary 3 c 2 5 d 2 p4 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 Chabot College Mathematics 33 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 C sketch and label the A B angles A, B, and C. Recall that the measures of the angles in any triangle add to 180°. Chabot College Mathematics 34 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Missing Term Translate: This geometric fact about triangles provides A + B + C = 180. one equation: Translate Relationship Statements Angle B is three times the measure of angle A. B = Chabot College Mathematics 35 3A Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Missing Term Translate Relationship Statements Angle C is 60o greater than twice the measure of A C = Translation Produces the 3-Equation System Chabot College Mathematics 36 60 + 2A A B C 180 B 3A C 60 2 A Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Missing Term Since this System has A B C 180 Missing Terms in two B 3A of the Equations, Substitution is faster C 60 2 A than Elimination Sub into Top Eqn A 3A 60 2 A 180 • B = 3A • C = 60+2A Chabot College Mathematics 37 60 6 A 180 6 A 120 A 20 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example Missing Term BackSub A = 20° into the other eqns B 3A 3(20) 60 C 60 2 A 60 2(20) 100 Check → 20° + 60° + 100° = 180° State: The angles in the triangle measure 20°, 60°, and 100° Chabot College Mathematics 38 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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. Chabot College Mathematics 39 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 40 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 41 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example CAT Scan CAT Scan Tissue-Type Ranges LAU Linear Attenuation Units Chabot College Mathematics 42 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 43 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 44 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example CAT Scan Thus the Equation System 0.80 x y z 0.55 x y z 0.65 (1) (2) (3) Even with Missing Terms Elimination is sometimes a good solution method x y 0.80 Add −1 times Equation (1) x z 0.55 (2) to Equation (2) y z 0.25 (4) Chabot College Mathematics 45 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example CAT Scan 0.80 (1) The Replacement x y Operation Produces y z 0.25 (4) the Equivalent System y z 0.65 (3) Add Equation (4) to Equation (3) to get 0.80 (1) x y y z 0.25 (4) 2z 0.40 (5) Chabot College Mathematics 46 0.40 z 2 Or z 0.20 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 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 Chabot College Mathematics 47 y 0.20 0.25 y 0.45 y 0.45 x y 0.80 x 0.45 0.80 x 0.35 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Example CAT Scan Summarizing Results Recall Tissue-Type Table x 0.35 y 0.45 z 0.20 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) Chabot College Mathematics 48 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt WhiteBoard Work Problems From §3.3 Exercise Set • 46 An Inconsistent System WHY? Chabot College Mathematics 49 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt All Done for Today Carl Friedrich Gauss Chabot College Mathematics 50 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt Chabot Mathematics Appendix r s r s r s 2 2 Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu – Chabot College Mathematics 51 Bruce Mayer, PE BMayer@ChabotCollege.edu • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt