Chapter 1: Solving Linear Equations Chapter 1: Solving Linear Equations Assignment Sheet Date Topic Course Introduction Tech Day 1.1-­‐ Solving Simple Equations 1.2.1-­‐ Solving Multi-­‐Step Equations 1.2.2-­‐ Solving Multi-­‐Step Equation Word Problems 1.3.1-­‐ Solving Equations with Variables on Both Sides 1.3.2-­‐ Distance, Mixture, and Work Problems 1.4-­‐ Solving Absolute Value Equations 1.5-­‐ Rewriting Equations and Formulas Review Review EXAM Assignment 1) Sign class syllabus, both student AND parent 2) Get supplies by Monday 3) Download suggested iPad apps Get supplies by Monday Completed Pg. 8 #1-­‐15 (odd), 21-­‐35 (odd), 41, 43 Pg. 16 #1-­‐13 (odd), 17-­‐23 (odd) Pg. 16 #15, 16, 29-­‐35 (odd), 43, 46 Pg 23 #3-­‐15 (odd), 19-­‐23 (odd) Distance, Mixture, and Work worksheet Pg. 32 #1, 2, 3-­‐9 (odd), 11-­‐
23 (odd), 27-­‐30, 35 Pg. 40 #1-­‐13 (odd), 17, 27-­‐
30, 32, 33 Pg. 44 #1-­‐9, 12-­‐23 Cumulative Review: choose 3 problems from each bold headed section NO HOMEWORK 1.1-­‐ Solving Simple Equations Expression-­‐ Examples: Equation-­‐ Examples: Linear Equation-­‐ Examples: Solution-­‐ Examples: Addition Property of Equality: ________________________ the same number on __________________________________________________ of an equation. Subtraction Property of Equality: ______________________________ the same number on _________________________________________________ of an equation. Multiplication Property of Equality: _________________________ by the same number on_________________________________________________ of an equation. Division Property of Equality: __________________________ the same number on ________________________________________ of an equation. ***Using any of these properties creates _________________________________________________. ***Use ___________________________ _______________________ to solve for a missing variable in linear equations. ***Using ________________________________________ also keeps the equations _________________ Examples: Solve each equation. Justify each step. Check your answer. 1) x − 3 = −5 !!
4) ! = −3 2) 0.9 = y + 2.8 5) 1.3𝑧 = 5.2 !
!
3) 𝑚 + ! = ! !
6) ! 𝑡 = 6 You try: Solve each equation. Justify each step. Check your answer. !
!
7) n + 3 = −7 8) 𝑔 − ! = ! 9) -­‐6.5 = p + 3.9 !
!
10) ! = −6 11) 0.05𝑤 = 1.4 12) − ! 𝑚 = 21 13) In the 2012 Olympics, Usain Bolt won the 200 meter dash with a time of 19.32 seconds. Write and solve an equation to find his average speed to the nearest hundredth of a meter per second. Use the distance formula 𝑑 = 𝑟𝑡 14) Suppose Usain Bolt ran the 400 meters at the same average speed that he ran the 200 meters. How long would it take him to run 400 meters? Round your answer to the nearest hundredth of a second. 15) On January 22, 1943, the temperature in Spearfish, South Dakota, fell from 54°F at 9:00 a.m. to − 4°F at 9:27 a.m. How many degrees did the temperature fall? Write and solve an equation. 16) You thought the balance in your checking account was $68. When your bank statement arrives, you realize that you forgot to record a check. The bank statement lists your balance as $26. Write and solve an equation to find the amount of the check that you forgot to record. Homework: pg. 8 #1-­‐15 odds, #21-­‐35 odds, #41, 43 1.2-­‐ Solving Multi-­‐Step Equations Some algebraic equations require multiple steps in order to solve for the missing variable. Example: Solve 2.5𝑥 − 13 = 2 Check your solution. You try: Solve for the variable using inverse operations. Check your solution. !
1. −2𝑛 + 3 = 9 2. −21 = ! 𝑐 − 11 Some multi-­‐step linear equations require you to simplify first. One way you can simplify expressions is by combining like terms. Combine like terms of the following: a. 2 + 4 + d b. 12x + 4 + 2x c. 9z – 5 – 4z + 2 Solve -­‐12 = 9x – 6x + 15 Check your solution. Another way you can simplify is by using the distributive property. Use the Distributive Property to simplify the following expressions: a. 8(y + 3) b. 2(3x – 5) Solve the following. Be sure to check your solutions. 1) 3(z + 7) = 21 2) 2(1 – x) + 3 = -­‐8 c. -­‐4(z – 1) You try: 1) −2𝑥 − 10𝑥 + 12 = 18 2) 3(x + 1) + 6 = -­‐9 3) 13 = −2 ( y − 4 ) + 3y 4) 2x ( 5 − 3) − 3x = 14 5) 5 ( 3 − t ) + 2 ( 3 − t ) = 14 6) 15 = 5 + 4 ( 2d − 3) Homework: pg. 16 #1-­‐13 (odd), 17-­‐23 (odd) 1.2-­‐ Solving Real-­‐Life Problems 1) Use the table to find the number of miles x you need to bike on Friday so that the mean number of miles biked per day is 5. You try: !
2) The formula 𝑑 = ! 𝑛 + 26 relates the nozzle pressure n (in pounds per square inch) of a fire hose and the maximum horizontal distance the water reaches d (in feet). How much pressure is needed to reach a fire 50 feet away? 3) Your school’s drama club charges $4 per person for admission to a play. The club borrowed $400 to pay for costumes and props. After paying back the loan, the club has a profit of $100. How many people attended the play? You try: 4) You have 96 feet of fencing to enclose a rectangular pen for your dog. To provide sufficient running space for your dog to exercise, the pen should be three times as long as it is wide. Find the dimensions of the pen. 5) You order two tacos and a salad. The salad costs $2.50. You pay 8% sales tax and leave a $3 tip. You pay a total of $13.80. How much does one taco cost? 6) The difference of three times a number and 4 is -­‐19. What is the number? Homework: pg. 16 #15, 16, 29-­‐35 (odd), 43, 46 1.3.1-­‐ Solving Equations with Variables on Both Sides To solve an equation with variables on both sides, use inverse operations to collect the variables on one side and the constant terms on the other side. Then solve the one-­‐step equation. Examples: Solve for the missing variable and check your solution. a. 3𝑘 + 45 = 8𝑘 + 25 b. 7 − 5𝑧 = 17 + 5𝑧 c. 10 − 4𝑥 = −9𝑥 !
d. 3 3𝑥 − 4 = ! (32𝑥 + 56) So far, all of the linear equations we have seen have had one solution. This is not always the case! Infinite Solutions-­‐ No Solution-­‐ Examples: Solve each equation. Determine whether the equation has one solution, no solution, or infinitely many solutions. e. 3 5𝑥 + 2 = 15𝑥 f. −2 4𝑦 + 1 = −8𝑦 − 2 You try: Solve each equation. Determine whether the equation has one solution, no solution, or infinitely many solutions. 1. −2𝑥 = 3𝑥 + 10 !
2. ! 6ℎ − 4 = −5ℎ + 1 3. 4 1 − 𝑝 = −4𝑝 + 4 !
4. 6𝑚 − 𝑚 = ! (6𝑚 − 10) 5. 10𝑘 + 7 = −3 − 10𝑘 Homework: pg 23 #3-­‐15 (odd), 19-­‐23 (odd) 1.3.2-­‐ Distance, Mixture, and Work Problems 1) A boat leaves New Orleans and travels upstream on the Mississippi River for 4 hours. The return trip takes only 2.8 hours because the boat travels 3 miles per hour faster downstream due to the current. How far does the boat travel upstream? Hint: Use the distance formula. 𝑑 = 𝑟𝑡 2) A boat travels upstream on the Mississippi River for 3.5 hours. The return trip takes only 2.5 hours because the boat travels 2 miles per hour faster downstream due to the current. How far does the boat travel upstream? Hint: Use the distance formula. 𝑑 = 𝑟𝑡 3) A passenger on plane makes a trip to Las Vegas and back. On the trip there it flew 432 mph and on the return trip it went 480 mph. How long did the trip there take if the return trip took nine hours? 4) 9 lbs. of mixed nuts containing 55% peanuts were mixed with 6 lbs. of another kind of mixed nuts containing 40% peanuts. What percent of the new mixture is peanuts? 5) Emily mixed together 9 gal. of Brand A fruit drink and 8 gal. of Brand B fruit drink which contains 48% fruit juice. Find the percent of fruit juice in Brand A if the mixture contained 30% fruit juice. 6) Shawna can pour a large concrete driveway in six hours. Dan can pour the same driveway in seven hours. Find how long it would take them if they worked together. Homework: Distance, Mixture, and Work worksheet Distance, Mixture, and Work Homework 1) Working together, Jenn and Pat can mop a warehouse in 5.14 hours. Had she done it alone it would have taken Jenn 12 hours. How long would it take Pat to do it alone? 2) 9 gal. of a sugar solution was mixed with 6 gal. of a 90% sugar solution to make an 84% sugar solution. Find the percent concentration of the first solution. 3) Max left the science museum and drove south. Gabriella left three hours later driving 42km/h faster in an effort to catch up to him. After two hours Gabriella finally caught up. Find Max’s average speed. 4) Working alone, Ryan can dig a 10 ft by 10 ft hole in five hours. Trevor can dig the same hole in six hours. How long would it take them if they worked together? 5) A metallurgist needs to make 12.4 lbs. of an alloy containing 50% gold. He is going to melt and combine one metal that is 60% gold with another metal that is 40% gold. How much of each should he use? 6) A submarine left Hawaii two hours before an aircraft carrier. The vessels traveled in opposite directions. The aircraft carrier traveled at 25 mph for nine hours. After this time the vessels were 280 miles apart. Find the submarine’s speed. 1.4-­‐ Solving Absolute Value Equations Absolute Value-­‐ Examples: To solve 𝑎𝑥 + 𝑏 = 𝑐 when c ≥ 0, solve the related linear equations ____________________________________________ or _________________________________________________ **When c < 0, the absolute value equation 𝑎𝑥 + 𝑏 = 𝑐 has _____________________________ because absolute value always indicates a number that is not negative. ***Absolute value equations can have one solution, two solutions, or no solution. Examples: Solve each equation, if possible. Check for extraneous solutions. a. 𝑥 = 10 b. 𝑥 − 4 = 6 You try: c. 3𝑥 + 12 = 0 d. 3𝑥 + 1 = −5 Before you can solve these examples, you must isolate the absolute value expression on one side of the equation. They must be in the form 𝑎𝑥 + 𝑏 = 𝑐 e. 3𝑥 + 9 − 10 = −4 f. 7𝑥 − 3 + 8 = 5 g. 4 2𝑥 + 7 = 16 You try: 1. 𝑥 − 1 = 4 3. 𝑥 − 2 + 5 = 9 2. 3 + 𝑥 = −3 4. 3 2𝑥 + 5 = 27 6. 5. −2 5𝑥 − 1 − 3 = −11 −9 + v
= 3 8
Homework: pg. 32 #1, 2, 3-­‐9 (odd), 11-­‐23 (odd), 27-­‐30, 35 1.5-­‐ Rewriting Equations and Formulas Literal Equation-­‐ an equation that has ________________________________ variables Examples: *You can rewrite literal equations in terms of one variable. Examples: Solve the literal equation for 𝑦 a. 𝑦 − 2𝑥 = 15 b. 3𝑥 − 𝑦 = −4 c. 3𝑦 + 4𝑥 = 9 d. 2𝑥 − 2𝑦 = 5 You try: Solve the literal equation for 𝑦 1) 𝑦 + 𝑥 = 11 2) 4𝑥 + 𝑦 = 2 3) 3𝑦 − 𝑥 = 9 4) 20 = 8𝑥 + 4𝑦 5) 5𝑥 − 2 = 8 + 5𝑦 Formula-­‐ a type of _______________________ that shows how one variable __________________ to another variable Examples: 𝑑 = 𝑟𝑡 𝐴 = 𝑙𝑤 !
𝐼 = 𝑃𝑟𝑡 𝐶 = ! (𝐹 − 32) 𝐼 = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡, 𝑃 = 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙, 𝑟 = 𝑟𝑎𝑡𝑒, 𝑡 = 𝑡𝑖𝑚𝑒 (𝑦𝑒𝑎𝑟𝑠) 𝐶 = 𝐶𝑒𝑙𝑠𝑖𝑢𝑠, 𝐹 = 𝐹𝑎ℎ𝑟𝑒𝑛ℎ𝑒𝑖𝑡 Examples: a. You own a rectangular lot that is 500 feet deep. It has an area of 100,000 square feet. To pay for a new water system, you are assessed $5.50 per foot of lot frontage. Let w represent the lot frontage. Find the lot frontage using the formula 𝐴 = 𝑙𝑤. b. You deposit $5000 in an account that earns simple interest. After 6 months, the account earns $162.50 in interest. What is the annual interest rate? c. A truck driver averages 60 miles per hour while delivering freight to a customer. On the return trip, the driver averages 50 miles per hour due to construction. The total driving time is 6.6 hours. How long does each trip take? You try: 1) How much money must you deposit in a simple interest account to earn $500 in interest in 5 years at 4% annual interest? 2) Which has the greater surface temperature: Mercury or Venus? Mercury’s surface temperature is 427°C and Venus’s surface temperature is 864°F. 3) A fever is generally considered to be a body temperature greater than 100°F. Your friend !
has a temperature of 37°C. Does your friend have a fever? Use the formula 𝐶 = ! (𝐹 − 32) Homework: pg. 40 #1-­‐13 (odd), 17, 27-­‐30, 32, 33