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- 1. LINEAR EQUATIONS Prepared By: Malik Sabah-ud-din http://basic2advanced.blogspot.comWeb Site: 1
- 2. ∗ Classify equations as linear, fractional, or rational, ∗ Solve linear equations, ∗ Solve equations leading to the form ax+b=0, and ∗ Solve application problems involving linear equations by developing mathematical models for real-life problems. GENERAL OBJECTIVE At the end of the lesson the students are expected to: http://basic2advanced.blogspot.comWeb Site: 2
- 3. ∗ Identify an equation, ∗ Classify equations as identity, conditional or equivalent, ∗ Distinguish a consistent from an inconsistent equation, ∗ Enumerate the properties of equality. TODAY’S OBJECTIVE At the end of the lesson the students are expected to: http://basic2advanced.blogspot.comWeb Site: 3
- 4. An equation is a statement that two mathematical expressions are equivalent or equal. DEFINITION EQUATION The values of the unknown that makes the equation true are called solutions or roots of the equation, and the process of finding the solution is called solving the equation. Example: 9x2 = 117x =+ x32x37 −=− 5x32x7x4 +++=+ 1 2x x 2x 3x + + = − +
- 5. ∗ An identity equation is an equation that is true for any number substituted to the variable. KINDS OF EQUATIONS 121)(x. 3)3(. 3443. 22 2 ++=+ −=− +=+ xxc xxxxb xxa Example: http://basic2advanced.blogspot.comWeb Site: 5
- 6. ∗ A conditional equation is an equation that is true only for certain values of the unknown. 12)3(x. 0124. 232. +=− =− −=+ xc xb xxa Example: Week 1 Day 1 http://basic2advanced.blogspot.comWeb Site: 6
- 7. ∗ Two equations with exactly the same solutions are called equivalent equations. 4. 2225. 205. = =+ = xc xb xa Example: The following are equivalent equations. Week 1 Day 1 http://basic2advanced.blogspot.comWeb Site: 7
- 8. ∗ An inconsistent equation is an equation that has no solution. ∗ A consistent equation is an equation that has a solution. Week 1 Day 1
- 9. EXAMPLE Determine whether the given equation is an identity or a conditional equation. ( )( ) ( ) 15 8 53 .5 1 3 1 3 1 .4 532 .3 96432x.2 339.1 22 2 xxx x x xx x xxx xx xxx =+ − − = − + − =+ ++=+ +−=− Week 1 Day 1
- 10. For all real numbers a , b and c 1. Addition Property of Equality If a = b then a + c = b + c 2. Subtraction Property of Equality If a = b then a – c = b – c 3. Multiplication Property of Equality If a = b then a ∙ c = b ∙ c 4. Division Property of Equality If a =b then 0cwhere c b c a ≠= PROPERTIES OF EQUALITY Week 1 Day 1
- 11. ∗ Define linear equations in one variable, ∗ Determine the difference between linear and nonlinear equations, ∗ Enumerate the steps in solving linear equations, ∗ Solve linear equations and equations involving fractions, ∗ Solve rational equations which are reducible to linear equations, ∗ Define extraneous solution. TODAY’S OBJECTIVE At the end of the lesson the students are expected to: Week 1 Day 2
- 12. ∗ An identity equation is an equation that is true for any number substituted to the variable. RECALL • An equation is a statement that two mathematical expressions are equivalent or equal. • A conditional equation is an equation that is true only for certain values of the unknown. • Two equations with exactly the same solutions are called equivalent equations. • An inconsistent equation is an equation that has no solution. • A consistent equation is an equation that has a solution. Week 1 Day 2
- 13. DEFINITION LINEAR EQUATION IN ONE VARIABLE A linear equation in one variable is an equation that can be written in the form ax + b = 0 where a and b are real numbers and a 0 Example: 2x – 1 = 0, -5x = 10 + x, 3x + 8 = 2 Week 1 Day 2
- 14. Linear Equations Nonlinear Equations 354 =−x 822 =+ xx 7 2 1 2 −= xx 06 =− xx 3 6 x x =− 12 3 =− x x Nonlinear; contains the square of the variable Nonlinear; contains the reciprocal of the variable Nonlinear; contains the square root of the variable Week 1 Day 2
- 15. SOLVING A LINEAR EQUATION IN ONE VARIABLE Steps 1. Simplify the algebraic expressions on both sides of the equation. 2. Gather all the variable terms on one side of the equation and all constant terms on the other side. 3. Isolate the variable. 4. Check the solution by substituting the value of the unknown into the original equation. Week 1 Day 2
- 16. EXAMPLE STEP DESCRIPTION EXAMPLE 1 Simplify the algebraic expression on both sides 2(x-1)+3 = x-3(x+1) 2x-2+3 = x-3x-3 2x+1 = -2x-3 2 Gather all the variables on one side of the equation and all constant terms on the other side. 2x+2x = -3-1 4x = -4 3 Isolate the variable 1-x 4 4 x = − = Problem #23 on page 97 Week 1 Day 2 Solve for the indicated variable: 2(x-1)+3=x-3(x+1) http://basic2advanced.blogspot.comWeb Site: 16
- 17. Solve the following equations. ( )[ ] ( ) ( )[ ]3y31y55y232y3-5y2-25 97.pp 32# +−−−−−=++ ( )[ ] ( ) ( )[ ]y643y2627y472-6y98y-7-46 97.pp 36# +−−−−−=+ Week 1 Day 2 http://basic2advanced.blogspot.comWeb Site: 17
- 18. Linear Equations Involving Fractions. 4 63 x2 7 x 97.pp 39# += 15 1x6 5 2x 3 5-x -1 97.pp 48# − − + = Week 1 Day 2
- 19. SOLVING RATIONAL EQUATIONS THAT ARE REDUCIBLE TO LINEAR EQUATIONS A rational equation is an equation that contains one or more rational expressions. Extraneous solution are solutions that satisfy a transformed equation but do not satisfy the original equation. Steps 1. Determine any excluded values(denominator equals 0). 2. Multiply the equation by the LCD. 3. Solve the resulting linear equation. 4. Eliminate any extraneous solution. Week 1 Day 2
- 20. a7 12 2 a 2 93.pp 1.1.4ex.Classroom .1 =− )4a(a 8 a 5 4-a 2 94.pp 1.1.5ex.Classroom .2 − =− x3x 1 6x2 1 12-4x 1 95.pp 1.1.6ex.Classroom .3 2 − = − − 3x 1 5-2x 2 95.pp 1.1.7ex.Classroom .4 + −= Solve the following equations. 4 2 1u u u Edition2ndWatsonandRedlin,by Stewart ryTrigonomet&Algebra 78page1.1exercise.5 = + − EXAMPLE Week 1 Day 2
- 21. ∗ Solve equations using radicals ∗ Solve absolute value equations ∗ Solve literal equations TODAY’S OBJECTIVE At the end of the lesson the students are expected to: Week 1 Day 3
- 22. ∗ Steps in solving linear equations ∗ A rational equation is an equation that contains one or more rational expressions. ∗ Steps in solving rational equations. ∗ Extraneous solution are solutions that satisfy a transformed equation but do not satisfy the original equation. RECALL Week 1 Day 3
- 23. SOLVING EQUATIONS USING RADICALS solutionrealnohasequationtheaandevenisnif aandevenisnifa oddisnifax solutionahasaxequationThe n n n ,0 0x < ≥±= = = existnotdoesbecausesolutionrealnohasx xsolutionrealoneonlyhasx xsolutionsrealtwohasx xsolutionrealoneonlyhasx Examples 1616 232:32 216:16 232:32 : 44 55 44 55 −−= −=−=−= ±=±== === Week 1 Day 3
- 24. 123x2.1 =− Solve each equations: Week 1 Day 3 51x3.2 =+ 126x25.3 =+−
- 25. ABSOLUTE VALUE EQUATIONS DEFINITION The absolute value of a number a is given by .linenumberrealtheonaandxbetweencetandistheisax ,generallyMore.originthetoaofcetandistherepresentsitthatand 0aifa 0aifa a − <− ≥ = Week 1 Day 3
- 26. Solve each equations: (examples on page 131) 1457x3.2 35x2.1 =+− =− Solve each equations: 2x31-x.17 1565x3.13 += =++ EXAMPLE Edition2ndWatsonandRedlin,by Stewart ryTrigonomet&Algebra 131pagefromExercise Edition2ndWatsonandRedlin,by Stewart ryTrigonomet&Algebra 133pagefromExercise Week 1 Day 3
- 27. SOLVING FOR ONE VARIABLE IN TERMS OF THE OTHER Many formulas in the sciences involve several variables, and it is often necessary to express one of the variables in terms of the others. 2 r r mM GF equationtheinMiablevatheforsolve = lhwhlwA equationtheinwiablevatheforsolve 222 r ++= Edition2ndWatsonandRedlin,by Stewart ryTrigonomet&Algebra 72-71pagefromExample Week 1 Day 3
- 28. ∗ LINEAR EQUATIONS ARE SOLVED BY : SUMMARY 1. Simplifying the algebraic expressions on both sides of the equation. 2. Gathering all the variable terms on one side of the equation and all constant terms on the other side. 3. Isolating the variable. 4. Checking the solution by substituting the value of the unknown into the original equation. • RATIONAL EQUATIONS ARE SOLVED BY : 1. Determining any excluded values(denominator equals 0). 2. Multiplying the equation by the LCD. 3. Solving the resulting linear equation. 4. Eliminating any extraneous solution. Week 1 Day 3
- 29. CLASSWORK HOMEWORK #s 31,33,35,43,46,51,55,57,61, 65 page 97-98 #s 32, 34, 42, 60 page 97 Week 1 Day 3
- 30. APPLICATION INVOLVING LINEAR EQUATIONS Week 2 Day 1 Application Involving Linear Equations (Algebra and Trigonometry, Young 2nd Edition, page 100-113).
- 31. ∗ Develop mathematical models for real-life problems, ∗ Solve application problems involving common formulas, ∗ Solve number problems, ∗ Solve digit problems, ∗ Solve geometric problems, and ∗ Solve money and coin problems. TODAY’S OBJECTIVE Week 2 Day 1
- 32. 1. Read and analyze the problem carefully and make sure you understand it. 2. Make a diagram or sketch, if possible. 3. Determine the unknown quantity. Choose a letter to represent it. 4. Set up an equation. Assign a variable to represent what you are asked to find. 5. Solve the equation for the unknown quantity. 6. Check the solution. STEPS IN SOLVING WORD PROBLEMS Week 2 Day 1
- 33. Start Read and analyze the problem Make a diagram or sketch if possible Determine the unknown quantity. Did you set up the equation? Set up an equation, assign variables to represent what you are asked to find. A no yes A Solve the equation Check the solution Is the unknown solved? no yes End Week 2 Day 1
- 34. 1. Find three consecutive odd integers so that the sum of the three integers is 5 less than 4 times the first. (Example 2 page 102) 2. Find two consecutive even integer s so that 18 times the smallest number is 2 more than 17 times the larger number. (Classroom Ex. 1.2.2 page 102) NUMBER PROBLEMS Week 2 Day 1
- 35. 1. A rectangle 3 inches wide has the same area as a square with 9 inch sides. What are the dimensions of the rectangle? (Your Turn problem page 103) 2. Consider two circles, a smaller one and a larger one. If the larger has a radius that is 3 feet larger than that of the smaller circle and the ratio of the circumferences is 2:1, what are the radii of the two circles. (#21 page 110) GEOMETRY PROBLEMS Week 2 Day 1
- 36. 1. In an integer between 10 and 100, the unit’s digit is 3 greater than the ten’s digit. Find the integer, if it is 4 times as large as the sum of its digits. (from Internet Guide to Engineering Mathematics) 2. A certain two digit number is equal to 9 times the sum of its digits. If 63 were subtracted from the number the digits would be reversed. Find the number. (from Internet Guide to Engineering Mathematics) 3. The sum of the digits of a two-digit number is 11. If we interchange the digits then the new number formed is 45 less than the original. Find the original number. (onlinemathlearning .com) DIGIT PROBLEMS Week 2 Day 1
- 37. 1. A change purse contains an equal number of pennies, nickels and dimes. The total value of the coins is $1.44. How many of each type does the purse contain? (# 25 page 89 Algebra and Trig. By Stewart, Redlin and Watson, 2nd edition) 2. Mary has $3.00 in nickels, dimes and quarters. If she has twice as many dimes as quarters and five more nickels than dimes, how many coins of each type doe she have? (# 26 page 89 Algebra and Trig. By Stewart, Redlin and Watson, 2nd edition) MONEY AND COIN PROBLEMS Week 2 Day 1
- 38. ∗ Solve investment problems, ∗ Solve age problems, and ∗ Solve mixture problems. TODAY’S OBJECTIVE At the end of the lesson the students are expected to: Week 2 Day 2
- 39. 1. An ambitious 14-year old has saved $1,800 from chores and odd jobs around the neighborhood. If he puts this money into a CD that pays a simple interest rate of 4% a year, how much money will he have in his CD at the end of 18 months? (Classroom Ex. 1.2.4 page 104) 2. Theresa earns a full athletic scholarship for college, and her parents have given her the $20,000 they had saved to pay for her college tuition. She decides to invest that money with an overall goal of earning 11% interest. She wants to put some the money in a low-risk investment that has been earning 8% a year and the rest of the money in a medium-risk investment that typically earns 12% a year. How much money should she put in each investment to reach her goal? (Example #5 page 105) INVESTMENT PROBLEMS Week 2 Day 2
- 40. 1. A father is four times as old as his daughter. In 6 years, he will be three times as old as she is now. How old is the daughter now? (# 22 page 89 Algebra and Trig. By Stewart, Redlin and Watson, 2nd edition) 2. A movie star, unwilling to give his age, posed the following riddle to a gossip columnist. “Seven years ago, I was eleven times as old ad my daughter. Now I am four times as old as she is.” How old is the star? (# 23 page 89 Algebra and Trig. By Stewart, Redlin and Watson, 2nd edition) AGE PROBLEMS Week 2 Day 2
- 41. 1. A mechanic is working on the coolant system of a vehicle with a capacity of 11.0 liters. Currently the system is filled with coolant that is 45% ethylene glycol. How much fluid must be drained and replaced with 100% ethylene glycol so that the system will be filled with coolant that is 60% ethylene glycol? (Classroom Ex. 1.2.6 page 106) 2. For a certain experiment, a student requires 100 ml of a solution that is 8% HCl(hydrochloric acid). The storeroom has only solutions that are 5% and 15% HCl. How many milliliters of each available solution should be mixed to get a 100 ml of 8% HCl? (# 33 page 111) MIXTURE PROBLEMS Week 2 Day 2
- 42. 3. A cylinder contains 50 liters of a 60% chemical solution. How much of this solution should be drained off and replaced with a 40% solution to obtain a final strength of 46%? (#30 page 37 Applied College Algebra and Trig. By Linda Davis 3rd edition) MIXTURE PROBLEMS Week 2 Day 2
- 43. ∗ Solve uniform motion problems, ∗ Solve work problems, and ∗ Solve clock problems. TODAY’S OBJECTIVE At the end of the lesson the students are expected to: Week 2 Day 3
- 44. 1. You and your roommate decided to take a road trip to the beach one weekend. You drove all the way to the beach at an average speed of 60 mph. Your roommate drove all ath e way back (on the same route, but with no traffic) at an average rate of 75mph. If the total trip drive took a total of 9 hours, how many miles was the trip to the beach? (Classroom Ex. 1.2.7 page 108) 2. A Cessna 150 averages 150 mph in still air. With a tailwind it is able to make a trip in 2 1/3 hours. Because of the headwind, it is only able to make a return trip in 3 ½ hours. What is the average wind speed? (Your turn problem page 108) UNIFORM MOTION PROBLEMS Week 2 Day 3
- 45. 3. A motorboat can maintain a constant speed of 16 mph relative to the water. The boat makes a trip upstream to a marina in 20 minutes. The return trip takes 15 minutes. What is the speed of the current? (# 41 page 111) 4. On a trip Jerry drove a steady speed for 3 hours. An accident slowed his speed by 30 mph for the last part of the trip. If the 190-mile trip took 4 hours, what was his speed during the first part of the trip? (#37 page 37 Applied College Algebra and Trig. By Linda Davis 3rd edition) UNIFORM MOTION PROBLEMS Week 2 Day 3
- 46. 1. Connie can clean her house in 2 hours. If Alvaro helps her, they can clean the house in 1 hour and 15 minutes together. How long would it take Alvaro to clean the house by himself? (Example #8 page 109) 2. Next-door neighbors Bob and Jim use hoses from both houses to fill Bob’s swimming pool. They know it takes 18 hours using both hoses. They also knew that Bob’s hose, used alone, takes 20% less time that Jim’s hose alone. How much time is required to fill the pool by each hose alone? (#48 page 91 Algebra and Trig. By Stewart, Redlin and Watson, 2nd edition) WORK PROBLEMS Week 2 Day 3
- 47. 3. It takes 7 people 12 hours to complete a job. If they worked at the same rate, how many people would it take to complete the job in 16 hours. (#22 page 37 Applied College Algebra and Trig. By Linda Davis 3rd edition) WORK PROBLEMS Week 2 Day 3
- 48. 1. What time after 8 o’ clock will the hands of the continuously driven clock be opposite each other? 2. What time after 5:00 am will the hands of the continuously driven clock extend in opposite direction? 3. What time after 3:00 pm will the hands of the continuously driven clock are together for the first time? 4. What time after 4 o’ clock will the hands of the continuously driven clock from a right angle? CLOCK PROBLEMS Week 2 Day 3
- 49. In real world many kinds of application problems can be solved through modeling with linear equations. The following procedure will help you develop the model. Some problems require development of a mathematical model, while others rely on common formulas. 1. Read and analyze the problem carefully and make sure you understand it. 2. Make a diagram or sketch, if possible. 3. Determine the unknown quantity. Choose a letter to represent it. 4.Set up an equation. Assign a variable to represent what you are asked to find. 5.Solve the equation for the unknown quantity. 6.Check the solution. SUMMARY Week 2 Day 3

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- Week 2 Day 1 Application Involving Linear Equations (Algebra and Trigonometry, Young 2nd Edition, page 100-113).
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