Chabot Mathematics § 3. 2 a System Applications Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@Chabot. College. edu Chabot College Mathematics 1 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Review § 3. 1 MTH 55 § Any QUESTIONS About • §’s 3. 1 → Systems of Linear Equations § Any QUESTIONS About Home. Work • §’s 3. 1 → HW-08 Chabot College Mathematics 2 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
System Methods Compared § We now have three distinctly different ways to solve a system. Each method has strengths & weaknesses Method Strengths Weaknesses 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. § Solutions are not graphically § Easy to use when fractions or displayed. decimals appear in the system. Chabot College Mathematics 3 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 4 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 5 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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) Chabot College Mathematics 6 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 7 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 8 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Problem Solving 1. Familarize with Diagram x y 2. Translate § § Since the angles are supplementary, one equation is x + y = 180 (1) The second sentence can be rephrased and translated as follows: Chabot College Mathematics 9 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Problem Solving 2. Translating § Rewording and Translating One angle is 20 more than three times the other y • 20 + 3 x (2) We now have a system of two equations and two unknowns. Chabot College Mathematics 10 = Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Problem Solving 3. Carry Out • Sub x = 40° in Eqn-1 Chabot College Mathematics 11 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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° Chabot College Mathematics 12 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Elimination Applications § Total-Value Problems § Mixture Problems Chabot College Mathematics 13 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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? Chabot College Mathematics 14 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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. Chabot College Mathematics 15 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Drinks Sold 1. Familiarize – LET: 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 Chabot College Mathematics 16 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Drinks Sold 2. Translate. • Translating & Rewording Income from small drinks Plus large drinks $2 s § + $3 l $2 Totals $190 = $190 Thus we have Constructed the System Chabot College Mathematics 17 $3 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 18 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 19 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Problem-Solving TIP § When solving a problem, see if it is patterned or modeled after a problem that you have already solved. Chabot College Mathematics 20 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 21 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Pizza Stones 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: LET • • c = the no. of circular stones r = the no. of rectangular stones Chabot College Mathematics 22 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Circular Rectangular Cost per pan $26 $34 Number of pans c r 26 c 34 r Money Paid Chabot College Mathematics 23 Total 37 $1138 c + r = 37 26 c + 34 r = 1138 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Pizza Stones 2. Translate – We have translated to a system of equations: c + r = 37 (1) 26 c + 34 r = 1138 (2) 3. Carry Out Multiply Eqn(1) by − 26 § Add Eqns (2)&(3): Chabot College Mathematics 24 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 25 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Mixture Problem § A Chemical Engineer wishes to mix a reagent that is 30% acid 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? Chabot College Mathematics 26 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Acid Mixxing 1. Familiarize. Make a drawing and then make a guess to gain familiarity with the problem • The Diagram t liters + 30% acid Chabot College Mathematics 27 f liters = 50% acid 20 liters 35% acid Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 10 L guess produced 8 L of pure-acid in the mix, but we need 0. 35(20) = 7 L of pure-acid in the mix Chabot College Mathematics 28 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 29 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Acid Mixxing § Pure-Acid Calculation Table § The Table Reveals a System of Equations Chabot College Mathematics 30 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Acid Mixxing 3. Carry Out: Solve Eqn System § Eliminate f by multiplying both sides – 0. 50 t – 0. 50 f = – 10 of equation (1) by − 0. 5 0. 30 t + 0. 50 f = 7 and adding them to the – 0. 20 t = – 3 corresponding sides of t = 15. equation (2): Chabot College Mathematics 31 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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. 30 t + 0. 50 f = 0. 30(15) + 0. 50(5) = 7 Chabot College Mathematics 32 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 $324. § How much does each twin make per hour? • i. e. ; What are the Wage RATES Chabot College Mathematics 33 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Wage Rate § In This Case LET: • E ≡ Ethan’s Wage Rate ($/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 Chabot College Mathematics 34 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 35 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Wage Rate § Carry Out Chabot College Mathematics 36 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 37 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 • • l ≡ the length w ≡ the width Chabot College Mathematics 38 w Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Geometry 2. Translate. § The Perimeter is 2 l + 2 w. The perimeter is 268 feet. 2 l + 2 w = 268 § The length is 34 ft more than the width l Chabot College Mathematics 39 = 34 + w Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Geometry § Now have a system of two equations and two unknowns. • • 2 l + 2 w = 268 l = 34 + w (1) (2) 3. Solve for w using Substitution Method Chabot College Mathematics 40 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 41 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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? Chabot College Mathematics 42 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 43 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Coffee Beans 2. Translate – Since a 60 -lb batch is being made, we have i + v = 60. • Present the information in a table. Italian Number of pounds i Venezuelan New Blend v 60 Price per pound $9. 95 $11. 25 Value of beans 9. 95 i Chabot College Mathematics 44 11. 25 v i + v = 60 $10. 50 630 9. 95 i + 11. 25 v = 630 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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). Chabot College Mathematics 45 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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, Chabot College Mathematics 46 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 47 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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 Chabot College Mathematics 48 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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? Chabot College Mathematics 49 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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. Chabot College Mathematics 50 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Simple Interest § TRANSLATE: LET • • § x ≡ amount invested in blue-chip bonds. y ≡ amount invested in tech bonds Tabulate Interest Income Invest P r t I = Prt Blue x 0. 12 1 0. 12 x Tech y 0. 08 1 0. 08 y Chabot College Mathematics 51 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Simple Interest § Translate TOTAL INTEREST statement Interest from Total + technology = Blue-chip Interest § Translate TOTAL Principal statement {Blu. Chip Prin. } plus {Tech Prin. } is $10 k Chabot College Mathematics 52 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Simple Interest § Carry Out: Solve Principal Eqn for y and then sub into Interest Eqn Chabot College Mathematics 53 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Simple Interest § Back Substitute x = 6500 into Principal Eqn to find y § Check Chabot College Mathematics 54 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Example Simple Interest § State: Ms. Jeung invested $3500 in technology Bonds and $6500 in blue-chip Bonds. Chabot College Mathematics 55 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
White. Board Work § Problems From § 3. 2 Exercise Set • 22 § Peppermint Patty Puzzled Chabot College Mathematics 56 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
All Done for Today Commercial Coffee Bean Blending Chabot College Mathematics 57 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@Chabot. College. edu – Chabot College Mathematics 58 Bruce Mayer, PE BMayer@Chabot. College. edu • MTH 55_Lec-10_sec_3 -1_2 Var_Lin. Sys_ppt
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