1. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §3.2a System Applications

2. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §’s3.1 → Systems of Linear Equations  Any QUESTIONS About HomeWork §’s3.1 → HW-08 3.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 3 Bruce Mayer, PE Chabot College Mathematics System Methods Compared  We now have three distinctly different ways to solve a system. Each method has strengths & weaknesses MethodStrengthsWeaknesses Graphical  Solutions are displayed visually. Works with any system that can be graphed.  Inexact when solutions involve numbers that are not integers or are very large and off the graph. Substitution  Always yields exact solutions.  Easy to use when a variable is alone on one side of an equation.  Introduces extensive computations with fractions when solving more complicated systems.  Solutions are not graphically displayed. Elimination  Always yields exact solutions.  Easy to use when fractions or decimals appear in the system.  Solutions are not graphically displayed.

4. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 4 Bruce Mayer, PE Chabot College Mathematics Solving Application Problems 1.Read the problem as many times as needed to understand it thoroughly. Pay close attention to the questions asked to help identify the quantity the variable(s) should represent. In other Words, FAMILIARIZE yourself with the intent of the problem Often times performing a GUESS & CHECK operation facilitates this Familiarization step

5. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 5 Bruce Mayer, PE Chabot College Mathematics Solving Application Problems 2.Assign a variable or variables to represent the quantity you are looking for, and, when necessary, express all other unknown quantities in terms of this variable. That is, Use at LET statement to clearly state the MEANING of all variables Frequently, it is helpful to draw a diagram to illustrate the problem or to set up a table to organize the information

6. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 6 Bruce Mayer, PE Chabot College Mathematics Solving Application Problems 3.Write an equation or equations that describe(s) the situation. That is, TRANSLATE the words into mathematical Equations 4.Solve the equation(s); i.e., CARRY OUT the mathematical operations to solve for the assigned Variables 5.CHECK the answer against the description of the original problem (not just the equation solved in step 4)

7. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 7 Bruce Mayer, PE Chabot College Mathematics Solving Application Problems 6.Answer the question asked in the problem. That is, make at STATEMENT in words that clearly addressed the original question posed in the problem description

8. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Problem Solving  Two angles are supplementary. One of the angles is 20° larger than three times the other. Find the two angles 1.Familarize Recall that two angles are supplementary if the sum of their measures is 180°. We could try and guess, but instead let’s make a drawing and translate. Let x and y represent the measures of the two angles

9. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Problem Solving 1.Familarize with Diagram yx 2.Translate  Since the angles are supplementary, one equation is x + y = 180 (1)  The second sentence can be rephrased and translated as follows:

10. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Problem Solving 2.Translating  Rewording and Translating We now have a system of two equations and two unknowns. One angle is 20  more than three times the other y = 20 + 3x (2)

11. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Problem Solving 3.Carry Out Sub x = 40° in Eqn-1

12. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Problem Solving 4.Check If one angle is 40° and the other is 140°, then the sum of the measures is 180°. Thus the angles are supplementary. If 20° is added to three times the smaller angle, we have 3(40°) + 20° = 140°, which is the measure of the other angle. The numbers check. 5.STATE One angle measures 40° and the other measures 140°

13. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 13 Bruce Mayer, PE Chabot College Mathematics Elimination Applications  Total-Value Problems  Mixture Problems

14. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 14 Bruce Mayer, PE Chabot College Mathematics Total Value Problems  EXAMPLE: Lupe sells concessions at a local sporting event. In one hour, she sells 72 drinks. The drink sizes are –small, which sells for $2 each –large, which sells for $3 each. If her total sales revenue was $190, how many of each size did she sell?

15. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Drinks Sold 1.Familiarize. Suppose (i.e., GUESS) that of the 72 drinks, 20 where small and 52 were large. The 72 drinks would then amount to a total of 20($2) + 52($3) = $196.  Although our guess is incorrect (but close), checking the guess has familiarized us with the problem.

16. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Drinks Sold LET 1.Familiarize – LET: s = the number of small drinks and l = the number of large drinks 2.Translate. Since a total of 72 drinks were sold, we must have s + l = 72. To find a second equation, we reword some information and focus on the income from the drinks sold

17. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Drinks Sold 2.Translate. Translating & Rewording Income from small drinks Plus Income from large drinks Totals$190 $2s + $3l= $190  Thus we have Constructed the System $2 $3

18. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Drinks Sold 3.Carry Out Solve (1) for l Use (3) to Sub for l in (2) Use Distributive Law Combine Terms Simplify to find s  Sub s = 26 in (3) to Find l

19. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Drinks Sold 4.Check: If Lupe sold 26 small and 46 large drinks, she would have sold 72 drinks, for a total of: 26($2) + 46($3) = $52 + $138 = $190 5.State: Lupe sold 26 small drinks 46 large drinks

20. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 20 Bruce Mayer, PE Chabot College Mathematics Problem-Solving TIP  When solving a problem, see if it is patterned or modeled after a problem that you have already solved.

21. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Problem Solving  A cookware consultant sells two sizes of pizza stones. The circular stone sells for $26 and the rectangular one sells for $34. In one month she sold 37 stones. If she made a total of $1138 from the sale of the pizza stones, how many of each size did she sell? Pizza Stone

22. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Pizza Stones LET 1.Familiarize – When faced with a new problem, it is often useful to compare it to a similar problem that you have already solved. Here instead of $2 and $3 drinks, we are counting $26 & $34 pizza stones. So LET: c = the no. of circular stones r = the no. of rectangular stones

23. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Pizza Stones cont.2 2.Translate – Since a total of 37 stones were sold, we have:c + r = 37 Tabulating the Data Can be Useful $113834r26c Money Paid 37rc Number of pans $34$26 Cost per pan TotalRectangularCircular c + r = 37 26c + 34r = 1138

24. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Pizza Stones 2.Translate – We have translated to a system of equations: c + r = 37(1) 26c + 34r = 1138(2) 3.Carry Out  Multiply Eqn(1) by −26  Add Eqns (2)&(3):

25. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Pizza Stones cont.4 3.Carry Out – Solve for r Find c using Eqn (1): 4.Check: If r = 22 and c = 15, a total of 37 stones were sold. The amount paid was 22($34) + 15($26) = $1138  5.State: The consultant sold 15 Circular and 22 rectangular pizza stones

26. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Mixture Problem  A Chemical Engineer wishes to mix a reagent that is 30% acid and another reagent that is 50% acid.  How many liters of each should be mixed to get 20 L of a solution that is 35% acid?

27. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Acid Mixxing 1.Familiarize. Make a drawing and then make a guess to gain familiarity with the problem The Diagram + 30% acid 50% acid 35% acid = t liters f liters 20 liters

28. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Acid Mixxing  To familiarize ourselves with this problem, guess that 10 liters of each are mixed. The resulting mixture will be the right size but we need to check the Pure-Acid Content:  Our 10L guess produced 8L of pure-acid in the mix, but we need 0.35(20) = 7L of pure-acid in the mix

29. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Acid Mixxing 2.Translate:  LET t = the number of liters of the 30% soln f = the number of liters of the 50% soln  Next Tabulate the calculation of the amount of pure-acid in each of the mixture components

30. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Acid Mixxing  Pure-Acid Calculation Table  The Table Reveals a System of Equations

31. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Acid Mixxing 3.Carry Out: Solve Eqn System  Eliminate f by multiplying both sides of equation (1) by −0.5 and adding them to the corresponding sides of equation (2): –0.20t = –3 0.30t + 0.50f = 7 –0.50t – 0.50f = –10 t = 15.

32. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 32 Bruce Mayer, PE Chabot College Mathematics Example  Acid Mixxing  To find f, we substitute 15 for t in equation (1) and then solve for f: 15 + f = 20 f = 5  Obtain soln (15, 5), or t = 15 and f = 5 4.Check: Recall t is the of liters of 30% soln and f is the of liters of 50% soln Number of liters: t + f = 15 + 5 = 20 Amount of Acid: 0.30t + 0.50f = 0.30(15) + 0.50(5) = 7 

33. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 33 Bruce Mayer, PE Chabot College Mathematics Example  Wage Rate  Ethan and Ian are twins. They have decided to save all of the money they earn at their part-time jobs to buy a car to share at college. One week, Ethan worked 8 hours and Ian worked 14 hours. Together they saved $256. The next week, Ethan worked 12 hours and Ian worked 16 hours and they earned $324.  How much does each twin make per hour? i.e.; What are the Wage RATES

34. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 34 Bruce Mayer, PE Chabot College Mathematics Example  Wage Rate  In This Case LET: E ≡ Ethan’s Wage Rate ($/hr) I ≡ Ian’s Wage Rate ($/hr)  Translate: Ethan worked 8 hours and Ian worked 14 hours. Together they saved $256 8∙{Ethan’s Rt} plus 14∙{Ian’s Rt} is $256

35. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  Wage Rate  Translate: Ethan worked 12 hours and Ian worked 16 hours and they earned $324. 12∙{Ethan’s Rt} plus 16∙{Ian’s Rt} is $324  Now have 2-Eqn System

36. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 36 Bruce Mayer, PE Chabot College Mathematics Example  Wage Rate  Carry Out

37. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  Wage Rate  Carry Out  Sub I = 12 into 1 st eqn to Find E  State Answer Ethan Earns $11 per hour Ian Earns $12 per hour

38. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 38 Bruce Mayer, PE Chabot College Mathematics Example  Geometry  The perimeter of a fence around the children’s section of the community park is 268 feet. The length is 34 feet longer than the width. Find the dimensions of the park. 1.Familiarize: Draw a Diagram and LET: l ≡ the length w ≡ the width l w

39. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 39 Bruce Mayer, PE Chabot College Mathematics Example  Geometry 2.Translate.  The Perimeter is 2l + 2w. The perimeter is 268 feet. 2l + 2w = 268  The length is 34 ft more than the width l = 34 + w

40. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 40 Bruce Mayer, PE Chabot College Mathematics Example  Geometry  Now have a system of two equations and two unknowns. 2l + 2w = 268 (1) l = 34 + w (2) 3.Solve for w using Substitution Method

41. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 41 Bruce Mayer, PE Chabot College Mathematics Example  Geometry  Sub 50 for w in one of the Orignal Eqns l = 34 + w = 34 + 50 = 84 feet 4.Check: If the length is 84 and the width is 50, then the length is 34 feet more than the width, and the perimeter is: 2(84) + 2(50), or 268 feet 5.State: The width is 50 feet and the length is 84 feet

42. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 42 Bruce Mayer, PE Chabot College Mathematics Example  Mixture Problem  A coffee shop is considering a new mixture of coffee beans. It will be created with Italian Roast beans costing $9.95 per pound and the Venezuelan Blend beans costing $11.25 per pound. The types will be mixed to form a 60-lb batch that sells for $10.50 per pound.  How many pounds of each type of bean should go into the blend?

43. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 43 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans 1.Familiarize – This problem is similar to one of the previous examples. Instead of pizza stones we have coffee beans We have two different prices per pound. Instead of knowing the total amount paid, we know the weight and price per pound of the new blend being made.  LET  LET: i ≡ no. lbs of Italian roast and v ≡ no. lbs of Venezuelan blend

44. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 44 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans 2.Translate – Since a 60-lb batch is being made, we have i + v = 60. Present the information in a table. ItalianVenezuelanNew Blend Number of pounds iv60 Price per pound $9.95$11.25$10.50 Value of beans 9.95i11.25v630 i + v = 60 9.95i + 11.25v = 630

45. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 45 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans cont.4 2.Translate - We have translated to a system of equations 3.Carry Out - When equation (1) is solved for v, we have: v = 60 − i. We then substitute for v in equation (2).

46. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 46 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans cont.5 3.Carry Out - Find v using v = 60 − i. 4.Check - If 34.6 lb of Italian Roast and 25.4 lb of Venezuelan Blend are mixed, a 60-lb blend will result. The value of 34.6 lb of Italian beans is 34.6($9.95), or $344.27. The value of 25.4 lb of Venezuelan Blend is 25.4($11.25), or $285.75,

47. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 47 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans cont.6 4.Check – cont. so the value of the blend is [$344.27 + $285.75] = $630.02. A 60-lb blend priced at $10.50 a pound is also worth $630, so our answer checks 5.State – The blend should be made from 34.6 pounds of Italian Roast beans 25.4 pounds of Venezuelan Blend beans

48. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 48 Bruce Mayer, PE Chabot College Mathematics Simple Interest  If a principal of P dollars is borrowed for a period of t years with interest rate r (expressed as a decimal) computed yearly, then the total interest paid at the end of t years is  Interest computed with this formula is called simple interest. When interest is computed yearly, the rate r is called an annual interest rate

49. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 49 Bruce Mayer, PE Chabot College Mathematics Example  Simple Interest  Ms. Jeung invests a total of $10,000 in Bonds from blue-chip and technology Companies. At the end of a year, the blue-chips returned 12% and the technology stocks returned 8% on the original investments.  How much was invested in each type of Bond if the total interest earned was $1060?

50. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 50 Bruce Mayer, PE Chabot College Mathematics Example  Simple Interest  Familiarize:  We are asked to find two amounts: that invested in blue-chip Bonds that invested in technology Bonds.  If we know how much was invested in blue-chip Bonds, then we know that the rest of the $10,000 was invested in technology Bonds.

51. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 51 Bruce Mayer, PE Chabot College Mathematics Example  Simple Interest  TRANSLATE: LET x ≡ amount invested in blue-chip bonds. y ≡ amount invested in tech bonds  Tabulate Interest Income InvestPrtI = Prt Bluex0.1210.12x Techy0.0810.08y

52. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 52 Bruce Mayer, PE Chabot College Mathematics Example  Simple Interest  Translate TOTAL INTEREST statement Interest from Blue-chip Interest from technology = Total Interest +  Translate TOTAL Principal statement {BluChip Prin.} plus {Tech Prin.} is $10k

53. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 53 Bruce Mayer, PE Chabot College Mathematics Example  Simple Interest  Carry Out: Solve Principal Eqn for y and then sub into Interest Eqn

54. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 54 Bruce Mayer, PE Chabot College Mathematics Example  Simple Interest  Back Substitute x = 6500 into Principal Eqn to find y  Check  

55. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 55 Bruce Mayer, PE Chabot College Mathematics Example  Simple Interest  State: Ms. Jeung invested $3500 in technology Bonds and $6500 in blue-chip Bonds.

56. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 56 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §3.2 Exercise Set 22  Peppermint Patty Puzzled

57. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 57 Bruce Mayer, PE Chabot College Mathematics All Done for Today Commercial Coffee Bean Blending

58. BMayer@ChabotCollege.edu MTH55_Lec-10_sec_3-1_2Var_LinSys_ppt 58 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –