Chapter77. Linear equations. Contents: A Linear equations B Rational equations C Problem solving D Mixture problems

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1 Chapter77 Linear equations Contents: A Linear equations B Rational equations C Problem solving D Miture problems

$12 LINEAR EQUATIONS (Chapter 7) Opening problem Mrs May set her class the following challenge: Find a fraction whose numerator is more than its denominator, and the value of the fraction is equal to$ b Can you use algebra to solve the problem? Many worded problems can be converted to symbols to make algebraic equations. By solving the equations, we can find solutions to the problems.

b Can you use algebra to solve the problem? Many worded problems can be converted to symbols to make algebraic equations. By solving the equations, we can find solutions to the problems.

2 12 LINEAR EQUATIONS (Chapter 7) Opening problem Mrs May set her class the following challenge: Find a fraction whose numerator is more than its denominator, and the value of the fraction is equal to 1. That s impossible! Stan said, If the numerator is larger than the denominator, the value of the fraction must be greater than 1! Things to think about: a Can you eplain why Stan is wrong? b Can you use algebra to solve the problem? Many worded problems can be converted to symbols to make algebraic equations. By solving the equations, we can find solutions to the problems. A LINEAR EQUATIONS Linear equations are equations in which the variable is raised only to the power 1. All linear equations can be written in the form a + b 0 where a and b are constants and a 0. Eamples of linear equations include +1, 2, and 10 7:y. ALGEBRAIC SOLUTION TO LINEAR EQUATIONS In previous years, we have seen how to solve linear equations algebraically: Step 1: Step 2: Step : Determine how the epression containing the unknown has been built up. Perform inverse operations on both sides of the equation to undo how the epression is built up. In this way we isolate the unknown. Check your solution by substitution. Eample 1 Solve for : a 17 b The inverse of a 17 is +. The inverse of ) fadding 1 to both sidesg is. ) 8 ) 8 fdividing both sides by g ) 2 Check: X

3 LINEAR EQUATIONS (Chapter 7) 127 b ) fsubtracting from both sidesg ) 1 ) 1 fdividing both sides by g The inverse of + is -. The inverse of is. ) 1 Check: ( 1 )+1 X EXERCISE 7A.1 1 Solve for : a +7 b 2 c 8 d 2 Solve for : a +921 b 2 7 c 1 d e +18 f g 12 8 h + Eample 2 Solve for : a 1 b 1 ( ) 1 a ) fadding to both sidesg ) 2 ) Check: 10 2 fmultiplying both sides by g ) X The inverse of is +. The inverse of is. 1 b ( ) 1 1 ) ( ) 1 fmultiplying both sides by g ) ) + + fadding to both sidesg ) 2 1 Check: (( 2) ) 1 1 X

4 128 LINEAR EQUATIONS (Chapter 7) Solve for : a 20 b 1 9 c 8 d 2 +7 e +2 1 f ( +7) g 1 h 1 7 ( ) 2 EQUATIONS WITH A REPEATED UNKNOWN In situations where the unknown appears more than once, we need to epand any brackets, collect like terms, and then solve the equation. To epand brackets we use the distributive law a(b+c) ab+ac. Eample Solve for : (2 ) 2( 1) (2 ) 2( 1) ) 2 + ( ) 2 2 ( 1) fepanding bracketsg ) ) 1 fcollecting like termsg ) fadding 1 to both sidesg ) 1 ) fdividing both sides by g Check: (2 ) 2( 1) 2 X If the unknown appears on both sides of the equation, we ² epand any brackets and collect like terms ² move the unknown to one side of the equation and the remaining terms to the other side ² solve the equation. Eample Solve for : a 2 + b (2 + ) a 2 + ) fsubtracting 2 from both sidesg ) ) ++ fadding to both sidesg ) 10 Check: LHS 10 2, RHS X

5 LINEAR EQUATIONS (Chapter 7) 129 b (2 + ) ) fepanding bracketsg ) 2 ) fadding to both sidesg ) 2 ) 2 fdividing both sides by g ) 1 2 which means 1 2 Check: LHS (2 + ( 1 2 )) , RHS 1 2. X EXERCISE 7A.2 1 Solve for : a b c ( 2) 2 18 d ( ) 21 e 2( +)+( +)1 f ( 2) + ( +1) 0 g (2 +1) 2( 2) 2 h ( ) + ( 1) i 2( ) ( ) 11 j (2 ) (7 2) 0 2 Solve for : a 2 +1 b c d +9 e 8 + f 72 g 2( +) + h ( +2) 1 i ( +) 7 j (2 ) +1 Solve for, and eplain your answer: a (2 ) + 10 b ( 2) (2 +1) Activity 1 Click on the icon to practice solving linear equations. LINEAR EQUATIONS B RATIONAL EQUATIONS Rational equations are equations involving fractions. To solve rational equations, we write all the fractions in the equation with the same lowest common denominator (LCD), and then equate the numerators. For fractions whose denominators involve the variable, the lowest common denominator is found in the same way as for numerical fractions.

6 10 LINEAR EQUATIONS (Chapter 7) For eample: ² in ² in ² in ² in the LCD is 12 the LCD is 9 the LCD is (2 1) (2 1) the LCD is ( + 1)( 2) Eample Solve for : µ ) has LCD 10 fto create a common denominatorg ) 2( + ) fequating numeratorsg ) +2 fepanding bracketsg ) fsubtracting 2 from both sidesg ) ) 2 fdividing both sides by g Notice the insertion of brackets here. EXERCISE 7B 1 Solve for : a 2 9 d +2 g b 1 e h c f i Eample Solve for : ) has LCD fto create a common denominatorg ) 1 fequating numeratorsg ) 1 fdividing both sides by g 2 Solve for : a 2 7 e 2 b 9 f c g d 8 7 h

7 LINEAR EQUATIONS (Chapter 7) 11 Eample 7 Solve for : 2 +1 ) µ µ has LCD ( ) fto create a common denominatorg ) (2 + 1) ( ) fequating numeratorsg ) 8 +9 fepanding the bracketsg ) fadding to both sidesg ) 11 fsubtracting from both sidesg ) 11 fdividing both sides by 11g Solve for : +1 a +2 2 d b e c f g h 2 2 i j 2 + k l Eample 8 Solve for : has LCD µ ) fto create a common denominatorg ) 2 (1 2) 2 fequating numeratorsg ) fepanding bracketsg ) 1 2 ) 2 fadding 1 to both sidesg ) 2 fdividing both sides by g Solve for : a 2 b 2 d e c 2 f

8 12 LINEAR EQUATIONS (Chapter 7) Solve for : a 8 2 c Activity e b + d + 2 f Solving equations Place the LHS and RHS epressions in the correct boes so that the resulting equations have the correct solutions shown. PRINTABLE WORKSHEET LHS Epressions + 1 ( ) ( +) ( 2) 8 RHS Epressions Solution C PROBLEM SOLVING Many problems can be translated into algebraic equations. To solve problems using algebra, we follow these steps: Step 1: Decide on the unknown quantity and allocate it a variable such as. Step 2: Translate the problem into an equation. Step : Solve the equation by isolating the variable. Step : Check that your solution satisfies the original problem. Step : Write your answer in sentence form, describing how the solution relates to the original problem.

LINEAR EQUATIONS (Chapter 7) 1 Eample 9 When a number is trebled and subtracted from 7, the result is 11. Find the number. Let be the number, so is the number trebled.

$When 18 is subtracted from 7, the result is 7 18 11. X Eample 10 EXERCISE 7C.1 What number must be added to both the numerator and the denominator of 1 to obtain a fraction equal to 7 8?$

9 LINEAR EQUATIONS (Chapter 7) 1 Eample 9 When a number is trebled and subtracted from 7, the result is 11. Find the number. Let be the number, so is the number trebled. When is subtracted from 7, we get 7. ) 7 11 ) 18 fsubtracting 7 from both sidesg ) fdividing both sides by g So, the number is. Check: trebled gives 18. When 18 is subtracted from 7, the result is X Eample 10 EXERCISE 7C.1 What number must be added to both the numerator and the denominator of 1 to obtain a fraction equal to 7 8? Let be the number. ) µ ) µ + + which has LCD 8( + ) fto obtain a common denominatorg ) 8(1 + ) 7( + ) fequating numeratorsg ) fepanding bracketsg ) fsubtracting 7 from both sidesg ) 1 So, the number is 1. 1 When four times a certain number is subtracted from 2, the result is. Find the number. 2 Two less than twice a certain number, is equal to more than four times the number. Find the number. I think of a number. If I divide the sum of 8 and the number by, the result is one more than one third of the number. Find the number. A plane flying from Brisbane to Sydney is carrying 0 rows of passengers, as well as 12 crew members. If there are 222 people on board the plane, and every seat is taken, how many passengers are in each row? During a volleyball training session, Keela drank twice as much water as Carol, and Xavier drank 100 ml more water than Keela. Between them they drank litres of water. How much water did Carol drink?

1 LINEAR EQUATIONS (Chapter 7) At a restaurant, Table A s bill of $72 is split equally between the people at the table.

$7 What number must be added to both the numerator and the denominator of 7 to obtain a fraction equal to 7 9?$ $8 What number must be subtracted from both the numerator and the denominator of to obtain a fraction equal to 1?$ Solve the challenge set by Mrs May. Discussion Consider this problem: Amber baked a batch of biscuits to take to her friend s house. They were still hot, so she left the lid off the container.

Solve the challenge set by Mrs May. Discussion Consider this problem: Amber baked a batch of biscuits to take to her friend s house. They were still hot, so she left the lid off the container.

10 1 LINEAR EQUATIONS (Chapter 7) At a restaurant, Table A s bill of $72 is split equally between the people at the table. Table B has two more people than Table A, and its bill of $108 is also split equally between the people at the table. Given that each person at Table A pays the same amount as each person at Table B, how many people were at Table A? 7 What number must be added to both the numerator and the denominator of 7 to obtain a fraction equal to 7 9? 8 What number must be subtracted from both the numerator and the denominator of to obtain a fraction equal to 1? 9 Write a fraction for which the sum of the numerator and denominator is 20, and the value of the fraction is equal to Consider the Opening Problem on page 12. Solve the challenge set by Mrs May. Discussion Consider this problem: Amber baked a batch of biscuits to take to her friend s house. They were still hot, so she left the lid off the container. On her way out the door she dropped the container and had to throw half of the biscuits away. Amber and her friend ate three quarters of what was left. Amber fed two biscuits to her friend s dog, and then there was only one left. How many biscuits did Amber bake? ² As a class, write a linear equation to describe this situation. ² Is there a more efficient way to solve this problem? USING A TABLE Some problems can be made easier to understand by placing the given information into a table. Eample 11 Sarah s father is now three times as old as Sarah. In 1 years time, Sarah s father will be twice as old as Sarah. How old is Sarah now? Let Sarah s present age be years, so her father s present age is years. Table of ages: So, +12( + 1) ) Now 1 years time ) Sarah +1 ) 1 Father +1 ) Sarah is currently 1 years old.

EXERCISE 7C.2 LINEAR EQUATIONS (Chapter 7) 1 1 Ellie is now four times as old as her son. In years time she will be three times as old as her son. How old is Ellie s son now?

In two years time, if his age is doubled it will match his brother s age. How old is Adrian now? Eample 12 Britney has only 2-cent and -cent stamps.

Type Number Value 2-cent 2 cents -cent +2 ( +2)cents ) 2 +( + 2) 178 fequating values in centsg ) 2 + + 10 178 ) 7 + 10 178 ) 7 18 ) 2 So, there are 2, 2-cent stamps.

A deli has an equal number of 00 ml and 1 litre cartons of milk, and double that number of 2 litre bottles of milk. If there are 8 L of milk in total, how many 00 ml cartons does the deli have?

11 EXERCISE 7C.2 LINEAR EQUATIONS (Chapter 7) 1 1 Ellie is now four times as old as her son. In years time she will be three times as old as her son. How old is Ellie s son now? 2 When George was born, his father was years old. George is now 10% of his father s age. How old is George now? Four years ago, Adrian was one quarter of his brother s age. In two years time, if his age is doubled it will match his brother s age. How old is Adrian now? Eample 12 Britney has only 2-cent and -cent stamps. Their total value is $1:78, and there are two more -cent stamps than there are 2-cent stamps. How many 2-cent stamps are there? Let be the number of 2-cent stamps. ) there are ( +2) -cent stamps. Type Number Value 2-cent 2 cents -cent +2 ( +2)cents ) 2 +( + 2) 178 fequating values in centsg ) ) ) 7 18 ) 2 So, there are 2, 2-cent stamps. Lana collects 20-cent and 0-cent coins. The total value of her coins is currently $9:70. If Lana has more 0-cent coins than 20-cent coins, how many of each type does she have? A deli has an equal number of 00 ml and 1 litre cartons of milk, and double that number of 2 litre bottles of milk. If there are 8 L of milk in total, how many 00 ml cartons does the deli have? Tickets to a concert cost $20, $, or $0 each. The number of $20 tickets sold was triple the number of $ tickets sold. 00 more $0 tickets were sold than $ tickets. If the tickets sold had a total value of $10 00, how many of each type of ticket was sold? 7 Olivia makes her own fruit and nut mi using dried fruit costing $8 per kilogram, and nuts costing $12 per kilogram. She made a total of 0 kg of fruit and nut mi costing $10:80 per kilogram. How many kilograms of each ingredient did she use?

1 LINEAR EQUATIONS (Chapter 7) Eample 1 I invest in oil shares which earn me 12% yearly, and in coal mining shares which earn me 10% yearly.

Let the amount I invest in coal mining shares be $.

0:1 +0:12 + 0 910 ) 0:22 + 0 910 ) 0:22 0 ) 0 0:22 Total 910 ) 200 ) I invest $200 in coal mining shares and $00 in oil shares.

12 1 LINEAR EQUATIONS (Chapter 7) Eample 1 I invest in oil shares which earn me 12% yearly, and in coal mining shares which earn me 10% yearly. If I invest $000 more in oil shares than in coal mining shares, and my total yearly earnings amount to $910, how much do I invest in each type of share? Let the amount I invest in coal mining shares be $. Type of shares Amount invested ($) Interest Earnings ($) Coal 10% 10% of Oil ( + 000) 12% 12% of ( + 000) From the earnings column of the table, 10% of + 12% of ( + 000) 910 ) 0:1 +0:12( + 000) 910 ) 0:1 +0: ) 0: ) 0:22 0 ) 0 0:22 Total 910 ) 200 ) I invest $200 in coal mining shares and $00 in oil shares. 8 Australian Airways shares pay a yearly return of 8%, while BankCorp shares pay 11%. Tom invests $100 more on BankCorp shares than on Australian Airways shares, and his total yearly earnings from both investments is $1020. How much did Tom invest in Australian Airways shares? 9 Inka invested three times as much money in mining shares as she invested in telecommunications shares. Mining shares pay a yearly return of 9%, and telecommunications shares earn 7% yearly. Inka s yearly income from the shares is $81. Find how much money Inka invested in each type of share. 10 Bruce collects stamps. He has si times as many 10-cent stamps as -cent stamps, and he has some 20-cent stamps as well. Bruce has 72 stamps with a total value of $8:0. How many of each stamp does he have? 11 Andy has invested in three companies: Baldish, Creweman, and Droners. Baldish shares pay 10% yearly, Creweman shares pay 9% yearly, and Droners shares pay % yearly. Andy has invested twice the amount of money in Creweman as in Baldish, and he has invested $0 000 in total. The yearly return from the share dividends is $2. How much did Andy invest in each company?

13 LINEAR EQUATIONS (Chapter 7) 17 D MIXTURE PROBLEMS The following problems are concerned with the concentration of a miture when one liquid is added to another. For eample, a % cordial miture contains % cordial and 9% water. If we add more cordial to the miture then it will become more concentrated. Alternatively, if we add more water then the miture will be diluted. Eample 1 How much water should be added to 2 litres of % cordial miture to produce a % cordial miture? Suppose we add litres of water to the miture. litres of water From the diagrams we can write an equation for the total amount of cordial in the miture: % of 2 L %of ( +2)L ) ( +2) 100 ) 10 ( +2) ) 10 ( + 2) fequating numeratorsg ) 10 + fepanding bracketsg ) ) litres of % (2+) litres of cordial miture % cordial miture ) 1 1 litres of water must be added to the miture. EXERCISE 7D 1 How much water must be added to 1 litre of % cordial miture to produce a % cordial miture? 2 How much water must be added to Lof10% methylated spirits to dilute it to a 7% methylated spirit miture? How much cordial must be added to 0 litres of % cordial miture to make a 10% cordial miture? How many litres of 12% weedkiller miture must be added to 2 litres of 7% weedkiller miture to make a 9% weedkiller miture? It is helpful to draw a diagram.

18 LINEAR EQUATIONS (Chapter 7) Dominic needs 1 litres of % saline solution. He has already made up % saline and 7% saline mitures. How much of each should he mi together?

a b Find the epressions for the volumes of water and % drug solution which should be mied together to form: i 1 ml of n% drug solution ii m ml of n% drug solution.

14 18 LINEAR EQUATIONS (Chapter 7) Dominic needs 1 litres of % saline solution. He has already made up % saline and 7% saline mitures. How much of each should he mi together? Yvonne needs m ml of an n% drug solution. The drug is stored dissolved in water as a % solution. a b Find the epressions for the volumes of water and % drug solution which should be mied together to form: i 1 ml of n% drug solution ii m ml of n% drug solution. For what values of n will the epressions in a be valid? Review set 7 1 Solve for : + a +7 1 b 2 2 Solve for : a 2( 7) (8 ) 9 b 11 +( 2) Four times the result of subtracting 1 from a certain number is equal to three times the number, add 2. Find the number. Solve for : a + Solve for : a b b What number must be subtracted from both the numerator and denominator of 9 to obtain a fraction equal to 8? 7 Three years ago, Evan was one third of his sister s age. In a year s time, Evan s age doubled will match his sister s age. How old is Evan now? 8 Elaine has 88 coins in a purse which are all -cent and 10-cent coins. The total value of the coins is $7. How many of each type of coin does Elaine have in her purse? 9 Simon invests twice the amount in petroleum shares that he does in technology shares. Petroleum shares pay a yearly return of 12%, while technology shares pay 7% per year. Simon s yearly income from these shares is $7. Find how much money Simon has invested in each type of share. 10 How many litres of 8% cordial miture must be added to litres of % cordial miture to make a % cordial miture?

$Solve for : a 2 Solve for : +9 a 2 1 b 7 8 b 7 2 + Write a fraction for which the sum of the numerator and denominator is, and the value of the fraction is.$

15 LINEAR EQUATIONS (Chapter 7) 19 Practice test 7A Click on the icon to obtain this printable test. Multiple Choice PRINTABLE TEST Practice test 7B Short response 1 Solve for : + a 7 b Solve for : a ( ) 9 + b ( 2) 2 +7 At a theme park, Setha went on twice as many rides as Chenda, who went on three more rides than Dara. Together, they went on 2 rides. How many rides did Dara go on? Solve for : a 2 Solve for : +9 a 2 1 b 7 8 b Write a fraction for which the sum of the numerator and denominator is, and the value of the fraction is. 7 When Amanda was born, her mother was 27 years old. Amanda is now one quarter of her mother s age. How old is Amanda now? 8 A supermarket sells tuna in 90 g and 200 g tins. In one day, 112 tins of tuna were sold, with a total mass of 18 kg. How many tins of each type were sold? 9 Edward makes his own lolly bags to give away at a party. He mies jelly fruits costing $9 per kilogram and chocolate buttons costing $12 per kilogram. How many kilograms of each lolly does Edward mi if he produces 1 kg of lolly bags costing $11:20 per kilogram? 10 How many litres of % disinfectant miture needs to be added to litres of % disinfectant miture to produce a % disinfectant miture?

10 LINEAR EQUATIONS (Chapter 7) Practice test 7C Etended response 1 Paula, Adam, and Jessica are siblings. Adam is half the age of Paula, and Jessica is one quarter of Paula s age.

2 Miriam has a stamp collection made up of 10-cent and 20-cent stamps. She has more 20-cent stamps than 10-cent stamps, and the stamps have a total value of $:0.

16 10 LINEAR EQUATIONS (Chapter 7) Practice test 7C Etended response 1 Paula, Adam, and Jessica are siblings. Adam is half the age of Paula, and Jessica is one quarter of Paula s age. In si years time, Jessica will be 7% of Adam s age. a How old is each sibling now? b How old will each sibling be in si years time? c In how many years time will Jessica be 7% of Paula s age? 2 Miriam has a stamp collection made up of 10-cent and 20-cent stamps. She has more 20-cent stamps than 10-cent stamps, and the stamps have a total value of $:0. a b Find the number of: i 10-cent stamps ii 20-cent stamps. Miriam adds 1 more stamps to her collection. Some are 20-cent stamps and others are 0-cent stamps. The new total value for the stamp collection is $9:90. Find the total number of: i 0-cent stamps ii 20-cent stamps. A grocer buys cashews for $9 per kilogram and peanuts for $:0 per kilogram. He wants to sell the nuts individually, and he also makes 12 kilograms of a peanut-cashew mi which costs him $ per kilogram. a How many kilograms of each nut does the grocer require to make the mi? b How many grams of each nut is in a 1 kg serve of the mi? c Over a one week period, the grocer sells 18 kg of nuts, which cost him $1:80. Find the total weight of each nut sold during the week. A laboratory assistant has an 8% ethanol miture. She needs to make a % ethanol miture for an eperiment. a How much water must be added to litres of 8% miture to produce a % ethanol miture? b After the eperiment there are 2 litres of % ethanol miture left over. How much pure ethanol does she need to add to increase the concentration to 8% ethanol? Ozshop shares pay a yearly return of %, while Flynn and Co shares pay 9%. Annie invests $200 more on Flynn and Co shares than on Ozshop shares. Her total yearly earnings from both investments is $97. a b How much money has Annie invested in: i Ozshop ii Flynn and Co? The net year, the yearly annual return of Ozshop has increased, and Annie s new return is $107. What is the new percentage return from Ozshop shares?

Section 8.1 Extra Practice

Section 8.1 Extra Practice Name: Section 8. Extra Practice Date:. BLM 8 6.. Solve each equation. Use a number line. a) c x b) 4 4. Solve each equation. Use models of your choice to represent the solutions. a) x 0.6 b) x. Solve each