Name: Date: Period: : Solving Equations and Word Problems Notes#9 Section 2.1 Solving Two-Step Equations Remember the Golden Rule of Algebra: Whatever you do to of an equation you must do to the. STEPS: 1. or constants to both sides of the equation. 2. or by the coefficient of the variable. Name the four properties of equality that you use to solve equations: Solve. Show your check step where indicated 1.) 9x + 6 = 51 2.) 8y 4 = 28 3.) 18 3x 57 (check) (check) (check) 4.) 4 8x = 12 5.)10 m 4 2 6.) x 15 12 9 1, Chapter 1
7.) x 15 12 8.) a 5 8 9.) 4 11 c 10.) 3.6m 2.4 15.6 11.) 6 2.5g 14 12.) 2 8 2 m 3 13.) 3 m 2 5 14.) 4 3 6 a 3 x 2 2 15.) 3 4 6 16.) y 2 5 17.) 3 z 1.2 8.6 18.) 2.3 t 3.1 2.4 1.5 2
For #19-20, define variables, write an equation, and solve. 19.) You are ordering tulip bulbs from a flower catalog. You have $14 to spend. The cost of 1 bulb is $0.75 plus $3.00 shipping. How many bulbs can you order? 20.) You have $20 to spend on music downloads. You found a website that charges $0.80 per song plus a $1.50 handling fee. How many songs can you download? For #21-22, solve and justify each step. k 21.) Solve 1 5 and justify each step. 12 22.) Solve 8 3y 14 and justify each step. k 1.) 1 5 1.) 12 2.) 4 12 k 2.) 3.) 48 k 3.) 3
Notes#10 Section 2.2: Solving Multi-Step Equations A. Review Recall the distributive property: 1.) Simplify. 10(b 12) 2.) Evaluate. (8n 1)3 for n 2 STEPS: 1. Clear or 2. Use the property to remove parentheses 3. Combine on each side 4. Add or subtract to isolate the variable 5. Multiply or divide to finish isolating the variable B. Combining like terms Solve. Show your check where indicated. 3.) 9x 4x = 20 4.) 7 4m 5 2m 1 5.) 2c 1 c 12 77 (check): C. Application Write an equation and solve. 6.) A carpenter is building a rectangular fence for a playground. One side of the fence playground is the wall of a building 70 ft wide. He plans to use 340 ft of fencing material. What is the length of the playground if the width is 70 ft? 4
D. Solving an equation with grouping symbols. ( FIRST!!) Solve. 7.) 2(3 + 4m) 9 = 45 8.) 24 2(2m +1) = -6 9.) 15 3(x 1) 9 E. Solving an equation that contains fractions or decimals To clear fractions, ALL terms by the To clear decimals, by,, etc. Solve. 10.) 2 x 5 5 11.) m 3 2 6 4 m 2 5 8 12.) 2 3 x 5 8 x 26 13.) 26.45 4.2 x 1.25 14.) 1.2x 3.6 0.3x 2.4 15.) 41.68 4.7 8.6y 5
Section 2.3: Equations with variables on both sides of the equal sign A. Review Simplify. 1.) 6x 2x 2.) 5x 5x 3.) 2x x 8x B. Variables on both sides Distribute, if necessary Combine like terms on each side Add/Subtract to get the variable terms on one side Add/Subtract to get the constant terms on the side Multiply/Divide to isolate the. Solve each equation. Show your check step where indicated. 4.) 6x 3 8x 21 5.) 7k 4 5k 16 (check): (check): 6.) 2( c 6) 9c 2 7.) 6d d 4 6
8.) 4(3 + 5y) 4 = 3 + 2(y 2) 9.) 3(m 5) + 1 = 2(m + 1) 9 C. Application 10.) A hairdresser is considering ordering a certain shampoo. Company A charges $4 per 8-ox bottle plus a $10 handling fee per order. Company B charges $3 per 8-ox bottle plus a $25 handling fee per order. How many bottles must the hairdresser buy to justify using Company B? 7
Notes#11 Section 2.3: Solving multi-step equations with strange answers When your answer looks like: 0 = 0, or 3 = 3, or -2x = -2x this means that this equation is. The answer is written as or When your answer looks like: 0 = 10, or 3 = -5 this means that this equation is. The answer is written as or Solve. 1.) 10 8a 2(5 4a) 2.) 6m 5 7m 7 m 3.) (4x 3) (2x 5) 7x 8 4.) 3( y 5) 2(4 y) (4y 16) ( y 7) Section 2.4: Ratio and Proportion A. Review Simplify. 1.) 135 180 2.) 35 25 40 14 B. Using unit rates Find each unit rate. 3.) $.72 per 16 oz 4.) $1.20 32 oz C. Application 5.) $24.60 2 dozen roses 8
6.) In 2004 Lance Armstrong won the Tour de France, completing the 3391-km course in about 83.6 hours. Find Lance s unit rate, which is his average speed. Write an equation that relates the distance he cycles d to the time t he cycles. Cycling at his average speed, about how long would it take Lance to cycle 185 km? D. Converting Rates Important Conversions: 5280 feet = 1 mile 60 min = 1 hour 60 sec = 1 min Convert each rate. 7.) A cheetah ran 300 feet in 2.92 seconds. What was the cheetah s average speed in miles per hour? 8.) A sloth travels 0.15 miles per hour. Convert this speed to feet per minute. 9.) The speed limit on the freeway is 65 miles per hour. Convert that to feet per second. E. Solving proportions Choose method: - Cross-Multiply (or butterfly). Be sure to distribute! OR - Multiply both sides by the LCD; cross-cancel. Solve. 10.) 3 12 11.) 1 x 12.) 5 y 2 5 m 7 4 6 9
13.) 52 m 14.) 2 6 15.) 105 r 4 5 7 c 168 8 16.) 8 21 a 42 17.) t 10 9 15 F. Using proprotions Solve using a proportion. 18.) A box of cereal weighing 350 grams contains 21 grams of fat. Find the number of grams of fat in the recommended serving size of 50 grams. 19.) 8 gallons of gas cost $26. If you only have $15 to spend, how much gas can you buy? G. Solving multi-step proportions Solve. 20.) x 2 14 x 10 21.) 3 w 6 5 w 4 10
22.) y 15 y 4 35 7 11
Notes#12 Section 2.5: Solving Word Problems: Perimeter, Integers, Distance Write an equation and solve each problem. Complete a table where appropriate. A. Perimeter problems 1.) The perimeter of a rectangle is 150 cm. The length is 15 cm greater than the width. Find the dimensions. 2.) The width of a rectangle is 2 cm less than its length. The perimeter of the rectangle is 16 cm. What is the length of the rectangle? B. Consecutive integer problems 3.) The sum of three consecutive integers is 126. What are the integers? 4.) The sum of three consecutive integers is 189. What are the integers? 12
C. D=RT: Same-direction travel (fast car distance) (slow car distance) = (distance apart) 5.) Two cars leave town at the same time heading in the same direction. One car travels at 60mph and the other travels at 40mph. After how many hours will they be 50 miles apart? Slow Car Distance Rate Time Fast Car 6.) Two trains leave the station at the same time heading in the same direction. One train travels at 42mph and the other travels at 35mph. After how many hours will they be 31.5 miles apart? Slow Car Distance Rate Time Fast Car 13
D. D=RT: Round-trip travel (fast car distance) = (slow car distance) 7.) Lisa drives into the city to buy a software program at a computer store. Because of traffic, she averages only 15 mi/h. On her drive home she averages 35 mi/h. If the total travel time is 2 hours, how long does it take her to drive to the store? To the comp. store Distance Rate Time Return home 8.) On his way to work from home, Bart averaged only 20 miles per hour. On his drive home, he averaged 40 miles per hour. If the total travel time was 1 ½ hours, how long did it take him to drive to work? To work Distance Rate Time Return home 14
E. D=RT: Opposite direction travel (fast car distance) + (slow car distance) = (distance apart) 9.) Two cars leave town at the same time going in opposite directions. One of them travels 60mph and the other travels at 30mph. In how many hours will they be 150 miles apart? Slow Car Distance Rate Time Fast Car 10.) Two trains leave the station at the same time going in opposite directions. One of them travels 40mph and the other travels at 50mph. In how many hours will they be 135 miles apart? Slow Train Distance Rate Time Fast Train 15
Notes#13 Section 2.5: Mixture Problems Mixture Problems Volume x %/$ = amount Volumes: x and (total) x Setup a table, write an equation, and solve each problem. A. Solving mixture problems (Mixture amount) x (unit cost) = (total cost) 1.) Raisins cost $2 per pound and nuts cost $5 per pound. How many pounds of each should you use to make a 30-lb mixture that costs $4 per pound? Raisins Nuts Mixture Cost of raisins + cost of nuts = total cost B. Solving percent mixture problems (Solution amount) x (concentration as a decimal) = (amount of acid) 2.) A chemist has one solution that is 40% acid and another solution that is 80% acid. How many liters of each solution does the chemist need to make 300 liters of a solution that is 64% acid? Solution 1 Solution 2 Final Solution acid in the 1 st solution + acid in 2 nd solution = acid in final solution 16
3.) A chemist has a 10% acid solution and a 60% acid solution. How many liters of each solution does the chemist need to make 200 L of a solution that is 50% acid? Solution 1 Solution 2 Final Solution 4.) A solution containing 30% insecticide is to be mixed with a solution containing 50% insecticide to make 200L of a solution containing 42% insecticide. How much of each solution should be used? Volume % Amount of insecticide Solution 1 Solution 2 Final Solution 17
5.) A solution containing 28% fungicide is to be mixed with a solution containing 40% fungicide to make 300L of a solution containing 36% fungicide. How much of each solution should be used? Volume % Amount of fungicide Solution 1 Solution 2 Final Solution 6.) The Nut Shoppe has 10kg of mixed cashews and pecans, which sell for $8.40 per kilogram. Cashews alone sell for $8 per kilogram, and pecans sell for $9 per kilogram. How many kilograms of each are in the mix? Weight in kg Price per kg Total price Cashews Pecans Mixture 18