Linear equations. Linear equations. Solving linear equations

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1 A B Solving problems using linear equations C Substitution and transposition in linear relations D Linear recursion relationships E Simultaneous equations F Solving problems using simultaneous equations Linear equations AREAS OF STUDY The solution of linear equations including literal linear equations Developing formulas from word descriptions, substitution of values into formulas Solution of worded problems involving linear equations Substitution and transposition in linear relations, such as temperature conversion The construction of tables of values from a given formula using technology Linear relations defined recursively and simple applications The algebraic and graphical solution of simultaneous linear equations with two variables Solution of worded problems involving simultaneous linear equations with two variables ebookplus A Digital doc 10 Quick Questions A linear equation is an equation which contains a pronumeral (unknown value) raised to the power of 1. Such an equation may also be called an equation of the fi rst degree. Eamples of linear or first degree equations include: 4 = 8, y = 7 1 and y = + 3. Equations of the type: y = 1, y =, 4 = 8, + y = 4 and y = 3 8 are not linear since they contain pronumerals which are raised to powers other than 1; in these cases, 1, 1,, and 3 respectively. A linear equation is an equation which contains a pronumeral raised to the power of 1. It may also be called an equation of the first degree. Solving linear equations When we are asked to solve an equation, we are to find the value of the pronumeral so that when it is substituted into the original equation, it will make the equation a true statement. Equations are solved by performing a number of inverse operations to both sides of the equation until the value of the unknown is found. Chapter 71

2 When solving equations, the order of operations process, BODMAS (that is, Brackets Of Division, Multiplication, Addition, Subtraction) is reversed. We may therefore apply the SAMDOB process (BODMAS in reverse). This means that the operations of subtraction and addition are taken care of first, followed by multiplication and division. Brackets are dealt with last. WORKED EXAMPLE 1 Solve the following equations. a 3 = 4 b 3 + += 8 c 10 = 6 THINK WRITE a 1 Write the given equation. a 3 = 4 (Optional step.) Rule up a table with two columns to the side of the equation. In the first column, note each of the operations performed on in the correct order. In the second column, write the corresponding inverse operation. The arrows indicate which operation to begin with. Operation Solve the equation by making the subject. Add 3 to both sides of the equation. 3 = = = 7 4 Divide both sides of the equation by. 7 = Simplify. = 3 1 (or 3.) b 1 Write the given equation. b 6 (Optional step.) Rule up a table with two columns. In the first column, note each of the operations performed on in the correct order. In the second column, write the corresponding inverse operation. The arrows indicate which operation to begin with. 3 Solve the equation by making the subject. Subtract from both sides of the equation. 6 += 8 Operation += =8 = Multiply both sides of the equation by 6. 6= Simplify. = 36 c 1 Write the given equation. c 10 3 = Inverse Inverse (Optional step.) As in a and b above. Operation Inverse Maths Quest 11 Standard General Mathematics for the Casio ClassPad

3 3 Solve the equation by making the subject. Subtract 10 from both sides of the equation. 4 Multiply both sides of the equation by. Divide both sides of the equation by = = 10 3 = 3 = = 3 = = 3 3 = Simplify. = Step is an optional step which may be used initially to help you become familiar with the process of solving equations. The answers may be checked by substituting the values obtained back into the original equation or using a calculator. If the pronumeral appears in the equation more than once, we must collect terms containing the unknown on one side of the equation and all other terms on the other side. WORKED EXAMPLE Solve for in the equation: 4 = THINK WRITE 1 Write the given equation. 4 = Transpose 4 to the LHS of the equation by subtracting it from both sides of the equation. 4 4 = = 6 3 Add 4 to both sides of the equation = = 10 4 Divide both sides of the equation by. 10 = Simplify. = 6 To check whether your answer is correct, substitute it back into the equation to see if it will make a true statement: LHS = ( ) 4 = 14 RHS = 4 ( ) + 6 = 14 as LHS = RHS, the solution is correct. If the equation contains brackets, they should be epanded first. In some cases, however, both sides of the equation can be divided by the coefficient in front of the brackets instead of epanding. WORKED EXAMPLE 3 Solve for : a ( + ) = 3( 6) b 4(6 + ) = 0. Chapter 73

4 THINK WRITE a 1 Write the given equation. a ( + ) = 3( 6) Epand each of the brackets on both sides of the equation. 3 Transpose 6 to the LHS of the equation by subtracting it from both sides of the equation. 4 Subtract 10 from both sides of the equation. Divide both sides of the equation by = = = = = = Simplify. = 7 b 1 Write the given equation. b 4(6 + ) = 0 On the Main screen, tap: Action Advanced solve Complete the entry line as: solve(4(6 + ) = 0,) Then press E. 3 Write the solution. Solving 4(6 + ) = 0 for gives = 1 If an equation contains a fraction, we should first remove the denominators by multiplying each term of the equation by the lowest common denominator (LCD). WORKED EXAMPLE 4 Find the value of which will make the following a true statement: + = 3. THINK WRITE 1 Write the given equation. + = 3 Determine the LCD of and 3. LCD of and 3 is 6. 3 Multiply each term of the equation by the LCD. ( + ) 6= = ( + ) 4 Simplify both sides of the equation. = ( + ) = Maths Quest 11 Standard General Mathematics for the Casio ClassPad

5 Epand the bracket on the LHS of the equation. + 4 = Add 3 to both sides of the equation = = 30 7 Subtract 4 from both sides of the equation = 30 4 = 6 8 Divide both sides of the equation by. 6 = 9 Simplify. = 1 (or.) Sometimes in equations containing fractions, a pronumeral appears in the denominator. Such equations are solved in the same manner as those in the previous eamples. However, care must be taken to identify the value (or values) for which the pronumeral will cause the denominator to be zero (0). If in the process of obtaining the solution the pronumeral is found to take such a value, it should be discarded. Even though the process of identifying the value of the pronumeral that causes the denominator to be zero is at this stage merely a precaution, this process should be practised as it will prove useful in future chapters. WORKED EXAMPLE For the equation = 1 : a state which value(s) of will cause the equation to be undefined b solve for. ebook plus s Tutorial int-087 Worked eample THINK a Identify the values of which will cause the denominator to be zero. Note: Once the equation has been solved, values which cause the denominator to be 0 will be discarded. WRITE b 1 Write the given equation. b = 1 On the Main screen, tap: Action Advanced solve Complete the entry line as: solve =, 1 Then press E. a First fraction: = 0 Second fraction: = 0 = 0 Third fraction: 1 = 0 = 1 cannot assume the values of 0 and 1, since this will cause the fraction to be undefined. Chapter 7

6 3 Write the solution. Note: The value of is not 0 or 1. Hence, it is a valid solution. Solving = 1 for gives = 7 = 1 REMEMBER A linear equation is an equation that contains a pronumeral raised to the power of 1. It may also be called an equation of the fi rst degree. are solved by using inverse operations. When solving linear equations the order of operations process, BODMAS, is reversed. If the pronumeral appears more than once, the terms containing the unknown are collected onto one side of the equation and the numbers onto the other. If the equation contains brackets, either epand, or divide both sides by the coefficient in front of the bracket. If an equation contains fractions, multiply each term of the equation by the LCD. To check your solution, substitute it back into the equation to see if it will make a true statement. EXERCISE A ebook plus s Digital doc SkillSHEET.1 Solving linear equations 1 WE 1 Solve the following equations. a 16 b 6 =.3 c + 6 = 4 d 7 = 9 e 3 + = f 3 = 10 g 0. = 10 h = 1 i = 1 3 j = 4 k 3 = l = 7 m 1 = n = 0 o 7 3= p + 3= = 7 q 8 7= r = = s = t = 13 u 3 + = v 6 = w 3 = = y 6 1 z 17 = 0 3 WE Solve for. a = b + = c 1 + = 7 d 1 17 = e = 6 f = 9 g + = h 3 = 3 i 1 = + 8 j 7 = + 1 k + = 3 6 l 1 + = m 8 3 = 4 n 1 + = o 13 3 = 4 6 p = 3 76 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

7 3 WE3 Solve for. a 4( 0) = 16 b 3(1 ) = 6 c ( + 8) = 0 d (3 4) = 6 e ( + 6) = 13 f ( 7) = 3 g 3( 7) = 4( + 3) h 8( + 1) = (7 3) i 4( + 3) = (7 4) + j ( 4) 3 + 7( ) = 0 4 WE4 For each of the following, find the value of which will make the statement true. a = b 3 = c =1 d + 3= e = f = g ( + ) + = h 7 3 ( 3 ) = i + = j = k 7 ( (3 3 6) 46 (6 ) ) 4 = l + 7) = 3 ( WE For each of the following: a state which value (or values) of will cause the equation to be undefined b solve for. 1 4 i + 1 = 3 4 ii 1 + = 3 (3 6) 1 4 iii + ( 1)( 1 )( + 1) 1 = 7 iv = v 3 (7 4) = vi 4 8 = 3 6 MC Without solving the equation = we know that will not be equal to: A 3 B 1 C 0 D 0 or 1 E 1 or 3 or 0 7 MC To solve 3 ( 1) 4 =, each term of the equation could be multiplied by: 3 A B 3 C 4 D E 6 8 MC In order to solve the equation = 4, the operations which must be performed are: 3 A both sides by, then by 3 B both sides by 3 C both sides by 3 D both sides by 3, then by 4 E both sides by 4, then by 9 Find the value of z, such that the solution to the following equation is = 1. 3 z = ( ) ) (+ 1) 10 Solve the following equation. 4 6 = 1 + ebookplus Digital doc SkillSHEET. Finding the lowest common multiple Chapter 77

8 B Solving problems using linear equations can often be used to help us in problem solving. This is usually done in the following way. 1. Identify the unknown and choose any convenient pronumeral (usually ) to represent it.. Use the information given in the problem to compose an equation in terms of the pronumeral. 3. Solve the equation to find the value of the pronumeral. 4. Interpret your result by relating the answer back to the problem. WORKED EXAMPLE 6 If the sum of twice a certain number and is multiplied by 3 and then divided by 7, the result is 9. Find the number. THINK WRITE 1 Assign the pronumeral to the unknown value. Let = the unknown number. Build the equation according to the information given. (a) Twice the number; this means, so write this. (b) The sum of twice the number and ; this means +, so add this on. (c) The sum is multiplied by 3; this means 3( + ). Add this on. Note: We include brackets to indicate the order of operations. (d) The result is divided by 7; this means 3 ( + ). Add this on. 7 (e) The result is 9; which means that all of the previous computations will equal 9. Write this. 3 Solve for. (a) Multiply both sides of the equation by 7. (b) Divide both sides of the equation by 3 since they are both divisible by 3. (c) Subtract from both sides of the equation. + 3( + ) 3 ( + ) 7 3 ( + ) = ( + ) 7= ( + ) = 63 3 ( + ) 63 = = 1 + = 1 = 16 (d) Divide both sides of the equation by. 16 = (e) Simplify. = 8 4 Answer the question. The unknown number is 8. Your answer can be easily verified by checking whether it will satisfy the conditions specified in the problem. 78 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

9 In this instance: the sum of twice a certain number and : multiplied by 3 : divided by 7 : the result is 9 : 8 = = = 9 9 = 9 (True) Therefore, the answer is correct. WORKED EXAMPLE 7 Sometimes the problem contains more than one unknown. In such cases one of the unknowns is called and the other unknown/s are then epressed in terms of. Sarah is buying tulip bulbs. Red tulip bulbs cost $.0 each, while yellow tulip bulbs cost $4.70 each. If bulbs cost Sarah $107.40, how many of each type did she buy? THINK 1 Define the variables. Note: Since there are bulbs altogether, then the number of yellow tulip bulbs is the number of red tulip bulbs; that is,. Write an epression for the cost of the red tulips. Note: 1 red tulip costs $.0; therefore red tulips cost.0. 3 Write an epression for the cost of the yellow tulips. Note: 1 yellow tulip costs $4.70; therefore tulips cost 4.70 ( ). WRITE Let = the number of red tulip bulbs. Let = the number of yellow tulip bulbs. Total cost of red tulip bulbs =.0 =. Total cost of yellow tulip bulbs = 4.70 ( ) = 4.7( ) Chapter 79

10 4 Formulate an equation relating the total cost of the red and yellow tulips and the epressions obtained in steps and 3. Solve the equation. (a) Epand the brackets on the LHS of the equation. (b) Collect the like terms on the LHS of the equation. The total cost of the red and yellow tulip bulbs is $ Also the total cost of red and yellow tulip bulbs is ( ). Therefore ( ) = = = (c) Subtract from both sides of the equation = = 4 (d) Divide both sides of the equation by = (e) Simplify. = 8 6 Interpret the answer obtained. There are 8 red and 14 (that is, 8) yellow tulip bulbs. 7 Answer the question. Sarah bought 8 red and 14 yellow tulip bulbs. WORKED EXAMPLE 8 A train (denoted as train 1) leaves station A and moves in the direction of station B with an average speed of 60 km/h. Half an hour later another train (denoted as train ) leaves station A and moves in the direction of the first train with an average speed of 70 km/h. Find: a the time needed for the second train to catch up with the first train b the distance of both trains from station A at that time. ebook plus s Tutorial int-088 Worked eample 8 THINK WRITE 1 Define the variables. Note: Since the first train left half an hour earlier, the time taken for it to reach the meeting point will be Write the speed of each train. Train 1: v 1 = 60 Train : v = 70 3 Write the distance travelled by each of the trains from station A to the point of the meeting. (Distance = speed time) 4 Equate the two epressions for distance. Note: When the second train catches up with the first train, they are the same distance from station A that is, d 1 = d. Solve for. (a) Epand the brackets on the LHS of the equation. Let = the time taken for train to reach train 1. Therefore the travelling time, t, for each train is: Train 1: t 1 = + 0. Train : t = Train 1: d 1 = 60( + 0.) Train : d = 70 When the second train catches up with the first train, d 1 = d 60( + 0.) = = Maths Quest 11 Standard General Mathematics for the Casio ClassPad

11 (b) Collect the like terms on the LHS of the equation by subtracting 70 from both sides of the equation. (c) Subtract 30 from both sides of the equation. (d) Divide both sides of the equation by = = = = = 10 (e) Simplify. = 3 6 Substitute 3 in place of into either of the two epressions for distance, say into d Substitute = 3 into d = 70 d = Evaluate. = 10 8 Answer the questions. a The second train will catch up with the first train in 3 hours after leaving station A. b Both trains will be 10 km from station A. REMEMBER To solve worded problems using linear equations, follow these steps: 1. Identify the variables.. Set up an equation by transforming the written information into an algebraic statement or statements. 3. Solve the equation. 4. Interpret the result by relating the answer back to the original problem. EXERCISE B Solving problems using linear equations 1 WE6 A number is multiplied by and then divided by 3, and the result is 3. Find the number. The average of three consecutive odd numbers is 3. Find the largest number. 3 Half of a certain number is subtracted from 6 and the result is then tripled, and the answer is 18. Find the number. 4 The sum of twice the number and 6 is 0. Find the number. Double the sum of a number and 3 is 18. Find the number. 6 The sum of one-third of a number and is 7. Find the number. 7 Fiona is buying tulip bulbs. Red tulip bulbs cost $6.40 each, while yellow tulip bulbs cost $.0 each. If 8 bulbs cost Fiona $167.0, how many of each type did she buy? 8 A rectangle is. times as long as it is wide. Find the dimensions of the rectangle if its perimeter is 6 cm. 9 In an isosceles triangle, sides of equal length are together 8 cm longer than the third side. If the perimeter of the triangle is 3 cm, what is the length of each side? Chapter 81

12 10 In a scalene triangle the first angle is 3 times as large as the second, while the third angle is 0 smaller than the second. Find the size of each angle; hence, name the triangle according to its angles sizes. 11 The price of coffee rose by 0% and the cost is shown on the coffee jar on the right. Find the cost of the same jar of coffee before the price rise. 1 All items at a clothing store have been reduced by 1%. If Stephanie purchased a shirt at the reduced price of $84.1, what was its original price? 13 MC a If 7 times a number subtracted from gives 3, then the number is: A 7 B 7 C 8 D 6 E b The sum of one-quarter of a number and 10 is 1. The value of the number is: A 100 B 0 C 40 D 0 E 0 14 a I am 3 times as old as my cousin Carla, who is 3 1 times as old as my daughter Nina. If our 3 total ages are 43 years, how old is my cousin? b Another cousin, Zara, is Carla s older sister. Zara is as many times as old as my daughter Nina as the number of years that she is older than Carla. How old is my other cousin? 1 Simon is only 16 years old, but he has already lived in 4 different countries because of his father s job. He was born and spent a few years of his early childhood in the USA, then the family moved to Germany where he stayed one year longer than he had in the USA. After that, he lived in London for twice as long as he had in Germany. Finally they came to live in Melbourne. So far, he has been in Australia for years less than he lived in America. a At what age did Simon leave his country of birth? b For how long did Simon live in each country? 16 To write up my History report, I borrowed some books from the school library. However, it soon became apparent that I did not have enough books. So I returned to the library and borrowed 3 times as many books as I had before. After a while, I realised that I did not really need half of the books, so I returned them to the library and now I have only two more books than I borrowed in the beginning. How many books did I originally borrow from the library? 17 In the storeroom of a fruit shop there were boes of apples, one of golden delicious and the other of jonathans, which were to be sold at $.80 and $3.0/kg, respectively. The apples, however, were accidentally mied together and, instead of sorting them out, the owner decided to sell them as they were. So as not to make a loss, he sold the mied apples at $3.10/kg. How many kilograms of each type of apple were there if together they weighed 3 kg? 18 Before the beginning of the summer season a shopkeeper ordered shorts and T-shirts from the factory. Each pair of shorts cost him $7.0 and a T-shirt cost $.0. If he ordered 00 items altogether and the total of the bill came to $11.0, how many pairs of shorts and how many T-shirts did he order? $ Maths Quest 11 Standard General Mathematics for the Casio ClassPad

13 ebook plus s Digital doc WorkSHEET.1 19 WE8 Ale and Nat are going for a bike ride. Nat can ride at 10 km/h, while Ale can develop a maimum speed of 1 km/h if he needs to. Nat leaves home at 10 am, while Ale stays behind for 1 minutes and then sets out to catch up with Nat. When will Ale reach Nat, assuming that both of them are riding at their maimum speed? 0 Two towns are 16 km apart. A car and a bus start travelling towards each other from their respective towns. The car moves 10 km/h faster than the bus. If they pass each other in 1. h after they began their respective journeys, find the speed of each vehicle. 1 Samuel is paddling with a constant speed towards a certain place he has marked on his map. With the aid of a current (which has a speed of km/h) it takes him only 1 h 0 min to reach his destination. However, on the way back he has to paddle against the current and it then takes him 4 h to reach his starting point. Find Samuel s speed on the still water. One administrative assistant can type 1. times as fast as another. If they both work together, they can finish a certain job in 6 hours. However, if the slower one works by herself, she will need 1 hours to finish the same job. How many hours will the quicker assistant need to complete the job by herself? 3 Maya needs to renovate her house. She has enough money to pay a plumber for 8 days or a carpenter for 1 days. For how many days can she pay the tradesmen if they both work at the same time? If Maya s net pay cheque will come in weeks, can she afford to hire both specialists till then? 4 The price of coffee rose by 0% and is now $8.0 per jar. Find the cost of the same jar of coffee before the rise in price. In a particular school a number of VCE students obtained a tertiary entrance score higher than 99.4 and 1% more students obtained a score higher than 99.0, but lower than If there were 43 students whose tertiary entrance scores were above 99.0, how many of those obtained a score above 99.4? 6 Isabel had some money in the bank on a long term deposit at % simple interest p.a. She then needed some money for a new car ($1080 less than she had in her term deposit), so she decided to borrow from the bank at.9% simple interest p.a. In the end it turned out that the interest she obtained from her investment was eactly equal to the interest she had to pay on her loan. Find out how much Isabel s new car cost her. C Substitution and transposition in linear relations A formula is an equation or a rule that defines the relationship between two or more variables. If a formula describes a relationship between two variables, both of which are to the power of 1, and does not contain terms that include a product or quotient of those variables, then such a relation is said to be linear. Chapter 83

14 The graph which represents a linear relation is a straight line. That is how the term linear is derived. For eample: 4y 7 = 0 and y = are linear relations, while + y y = 3 or + y = 9 or y = 7 are not (as eplained previously). Linear relations are often found in practical situations. For instance, the formula for the circumference of a circle, C = πdπ, and the formula for the conversion of temperature from degrees Celsius to degrees Fahrenheit, F = 9 C + 3, both describe linear relations. If all but one value in the formula are known, then the value of the unknown variable is evaluated by substituting the known values into the formula. WORKED EXAMPLE 9 If the formula for the conversion of temperature from degrees Celsius ( C) to degrees Fahrenheit ( F) is given by F = 9 C + 3, find the value of F when: a C = 3 C b C = 10 C. c Answer parts a and b using a CAS calculator. THINK WRITE a 1 Write the formula for the conversion of temperature. Substitute 3 C in place of C into the formula. Note: 9 C = 9 C a F = 9 C + 3 If C = 3 C, F = Evaluate. = = 9 4 Answer the question and include the appropriate unit. b 1 Write the formula for the conversion of temperature. Since C = 10 C, substitute 10 C in place of C into the formula. 3 C is equivalent to 9 F. b F = 9 C + 3 F = Evaluate. = = 14 4 Answer the question and include the appropriate unit. c 1 On the Main screen, tap: c Action Advanced solve Complete the entry lines as: 9 solve f = c+ f c 3, = 3 9 solve f = c+ f c 3, = 10 Press E after each entry. 10 C is equivalent to 14 F. 84 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

15 Write the solutions. 3 C is equivalent to 9 F 10 C is equivalent to 14 F In the previous eample, the same operations were performed for both values of C: first the required value of C was multiplied by 9 and then 3 was added. If we were asked to find the values in F for 0 other values of C, we would go through eactly the same procedure. To repeat this procedure 0 times would be rather boring and time consuming. In situations like this, it is much easier to create a table of values using a calculator or a spreadsheet instead of performing repetitive calculations by hand. WORKED EXAMPLE 10 Marsha wishes to place some money in a term deposit with an interest rate of.% p.a. for a -year period, but first she wants to know how much interest she will earn on her investment if she deposits between $000 and $3000. The amount of simple interest that she can earn is given by the formula I = PRT, where P is the principal (the amount she invests), R is the rate of interest per annum (p.a.), and T is the period of time, in years, for which the principal is to be invested. Use a spreadsheet to find the values of interest for each value of P, where P increases from $000 to $3000 inclusive, in increments of $0. THINK WRITE 1 Write the formula of the interest equation. I = PRT Write the value of each given variable. P = principal, ranging from $000 to $3000 in increments of $0 (that is, 000, 00, 100 and so on) R =.% =. 100 = 0.0 T = 3 Substitute the known values into the formula. Note: The value for P was not substituted since it is constantly varying. I = P 0.0 = 0.11P 4 On the Spreadsheet screen: Label column A: pvalue Enter the values of P into column A, beginning from 000 in increments of 0. Press E after each entry. Label column B: ivalue Complete the entry line in cell B as: = 0.11 A Then press E. Highlight cells B to B and tap: Edit Fill Range OK Chapter 8

16 Copy the values of the principal and corresponding interest payable into the table. P I = 0.11P P I = 0.11P The interest payable for any principal can now be readily obtained from the table of values. WORKED EXAMPLE 11 Marsha decided to place $000 in a term deposit for 1 year. She knew the banks in her area offered interest rates ranging from % p.a. to 7% p.a. and wanted to find out how much interest her investment would earn in each case. If the relation between the interest (I ) and rate of interest (R ) is given by I = 0R : a draw a straight line graph to represent the relationship b use the graph to find the amount of interest earned when the rate is i 6% ii 6.% c use the graph to find the rate which will pay interest of $110. ebook plus s Tutorial int-089 Worked eample 11 THINK WRITE/DRAW a 1 Write the given formula. a I = 0R and R 7 Choose the two given R values to substitute into the given formula and determine the corresponding I values. Note: The etreme values of a given range are the most convenient values to use. Let R = and R = 7. When R =, I = 0 = 100 (, 100) When R = 7, I = 0 7 = 140 (7, 140) 3 State the points as coordinates. Two points on the graph are (, 100) and (7, 140). 4 Rule a set of aes and label them. Note: Graph paper must be used to obtain an accurate graph. The aes must be scaled appropriately in order for values to be read off the graph. I ($) (, 100) 0 (7, 140) 0 7 R (%) 86 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

17 b i 1 To find I when R = 6%, draw a vertical line from the point 6 on the R-ais until it intersects with the straight line. From the point of intersection with the graph, draw a horizontal line until it intersects with the I-ais. - Read the value from the I-ais. - b i I ($) R (%) Answer the question. When the interest rate is 6%, the interest payable is $10. ii 1 To find I when R = 6.%, draw a vertical line from the point 6. on the R-ais until it intersects with the straight line. From the point of intersection with the graph, draw a horizontal line until it intersects with the I-ais. - Read the value from the I-ais. - ii I ($) R (%) Answer the question. When the interest rate is 6.%, the interest payable is $130. c 1 To find R when I = $110, draw a horizontal line from the point 110 on the I-ais - until it intersects with the straight line. From the point of intersection with the graph, draw a vertical line until it intersects with the R-ais. Read the value from the R-ais. c I ($) R (%) Answer the question. When the interest payable is $110, the rate of interest is.%. As shown in the previous eample for the relation I = 0R, we were able to obtain values of I, given R, and also obtain values of R, given I. Alternatively, this could have also been solved algebraically. For instance, in worked eample 11, part c, using the relation I = 0R, substitute I = 110 into the equation and solve for R: I = 0R 110 = 0R 110 0R = 0 0. = R R =.% However, if we need to find many values of R given various values of I, it would be more convenient to have the corresponding formula the formula which would have R on one side and everything else on the other side of the equal sign. The variable which is by itself is called the subject of the formula (that is, a formula describes its subject in terms of all other variables). In the formula I = 0R, I is the subject. So now our Chapter 87

18 task is to make R the subject. Thus, we need to rearrange our formula. Such a rearrangement is called transposition. To transpose the equation I = 0R I 0R divide both sides of the equation by 0. = 0 0 I R 0 = I R = 0 To rearrange or transpose the formula, we need to perform the same inverse operations to both sides of the equation until the desired result is achieved. WORKED EXAMPLE 1 a Transpose the equation 4 = y 3 to make y the subject. b Transpose the equation ( 3 ) = 4(y + ) to make the subject. THINK WRITE a 1 Write the given equation. a 4 = y 3 Add 3 to both sides of the equation = y = y 3 Divide each term on both sides of the 4 3 y equation by. + = 4 3 Simplify both sides of the equation. + = y y = + b 1 Write the given equation. b ( 3) = 4(y + ) To make the subject, on the Main screen, tap: Action Advanced solve Complete the entry line as: solve(( 3) = 4(y + ),) Then press E. 3 3 Write the solution. Solving ( 3) = 4(y + ) for gives y = (y + ). 1 To transpose the above equations, we use the same methods as those employed for solving linear equations. The only difference is that in the end we do not obtain a unique (or specific) numerical value for the required variable, but rather an epression in terms of other variables. 88 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

19 REMEMBER 1. If all variables but one are known in the formula, the value of the unknown variable may be found by substituting the known values into the formula. Prior to substitution, transpose the formula to make the unknown variable the subject.. Transposition is the rearrangement of the formula. It involves using the same methods as those employed for solving equations. 3. The subject of the formula is the variable which is by itself, on one side of the equation, while all other variables are on the other side. EXERCISE C Substitution and transposition in linear relations 1 WE9 Use the formula for conversion of temperatures, F = 9 C + 3, (where F represents degrees Fahrenheit and C represents degrees Celsius) to find F when C is equal to: a 3 b 100 c 1 d 0 e 19 f 3 g 7 h 0 The radius of the smallest circle of the target is. cm and the distances between the subsequent circles are cm. a Find the radius of the second, third, fourth and fifth circles. b If the circumference of a circle is given by the formula C = πdπ, find the circumference of each of the circles of the target (use π = 3.14). 3 WE 10 Currency rates fluctuate daily. On a certain day, 1 Australian dollar (A ) was equivalent to of cm 1 American dollar (US). Hence, on that particular day the relationship between two currencies US could be described as A =. Using this formula, set up a reference table for the foreign echange clerk, containing amounts between $ and $100 in increments of $ in both American and Australian currencies. (Use a spreadsheet or a CAS calculator to help you.). cm cm cm Chapter 89

20 4 A bank clerk, who deals with term deposits, wishes to have a reference table with the initial investments and corresponding values of investments at the end of the term, which she can refer to while advising her clients. With the current special interest rate at 4.6% p.a. for a 6-month term, the formula which could be used to set up such a table is A= P+ P , where P is the principal (the size of the investment) and A is the total 100 amount a client will receive at the end of 6 months. This special interest rate is offered for principal amounts between $000 and $ inclusive. Use the spreadsheet to set up the table of values in increments of $00. WE 11 A family travelled by car from one town to another. The towns are 70 km apart and the whole trip took them hours. Assuming that the speed was constant throughout the journey, the relationship between the time, t, and the distance travelled, d, could be modelled by the following equation: d = 4t. a On a graph, what would the horizontal ais show time or distance? b Draw a graph to represent the relationship. c Use the graph to estimate the distance travelled in i 1. h ii 3 h iii 4 h. d Use the graph to estimate how much time it took to travel i km ii 00 km iii 0 km. e How long had the family been travelling by the time they were 0 km away from their destination? 6 The graph at right shows the cost of production, C, of two-way mirrors at a certain factory during each week of operation. As can be seen from the graph, the cost, C, depends on the number of mirrors, n, produced that week. a Use the graph to estimate the cost of producing: i 10 mirrors ii mirrors iii 40 mirrors. b Use the graph to estimate the number of mirrors produced, when the cost is: i $0 ii $7 iii $300. c Find the equation of the line. d Use the equation obtained in part c to verify your answers to parts a and b. 7 WE1 Transpose each of the following equations to make the pronumeral, indicated in brackets, the subject. (If two pronumerals are indicated, make a separate transposition for each.) a (0, 34) 0 Number of mirrors 3y = 6 (y) y + 4 = 0 (, y) Cost ($) C n c + y 6 = 0 (y) 3 4y + 1 = 0 (y) e 3a 14 (a) p = 3k (k) g 3 a 4 b (a, b) 10 3a = a b (a, b) i S = n 4 (n) a = 3b 0.c (c) ( a 3) k y = 3( ) () l = b (a) m (3 d) = 6(f + 4) (d, f) n 7 ( a 4 b) ( b a a) = (a, b) 3 4 o 3 a (b b+ 3a) + = 1 (a, b) p 3 6y 6 = (, y) Maths Quest 11 Standard General Mathematics for the Casio ClassPad

21 Questions 8 to 11 refer to the following information. A gardener charges a $40 fied fee for each visit plus $1 per hour of work. 8 MC If C represents the total cost of a visit and t the time he worked (in hours), which of the following graphs represents the above information? A C B t (4, 76) C C 40 D 1 0 C t (, 100) E 40 0 t 1 C 40 0 t 40 0 t 0 40 C 9 MC Which of the following represents the relationship between t and C? A C t = 0 B 1t + C = 40 C 1t + 40 C = 0 D t = 1C + 40 E t + C = 0 10 MC When the relationship between t and C is transposed to make t the subject, it is then written as: C + 40 C 1 A t = B 1t + 40 = C C t = 1 40 C 3 C D t = + E t = 3 11 MC If the total bill came to $79, for how long did the gardener work? A 3 h B 3 h 1 min C 3 h 30 min D 3 h 4 min E 4 h 1 The sum of the interior angles of a regular polygon is given by S = (n ) 180, where n is the number of sides. a Transpose the formula to make n the subject. b Use the appropriate formulas to complete the following table: Polygon Number of sides (n) Sum of interior angles (S) Triangle 3 Heagon 6 Dodecagon 1 Nonagon 9 Heptagon (If you use a graphics calculator, create two separate tables using the appropriate formulas and then copy the values into your book.) ebook plus s Digital doc WorkSHEET. Chapter 91

22 D Linear recursion relationships If a sequence of n numbers (terms) is such that each number after the first is obtained by following a certain rule, then the equation defining the relationship between any term (nth term) and the previous (n 1)th term is called a recurrence relation. If each term in the sequence is obtained by adding the same number to the previous term, it would then be a linear recurrence relation. A linear recurrence relation is defined by: t n = t n 1 + d, t 0 = a ebook plus s Interactivity int-0803 Linear recursion relationships In words, this means the first term in the sequence is a and then each consecutive term is obtained by adding d to the preceding term. WORKED EXAMPLE 13 Write the first terms of the recurrence relation n = n 1 +, 0 = 6. THINK WRITE 1 Write the given recurrence relation. n = n 1 +, 0 = 6 Substitute n = 1 into the recurrence relation. 1 = Substitute 6 in place of 0. = Evaluate. = 8 Substitute n = into the recurrence relation. = 1 + = Substitute 8 in place of 1. = Evaluate. = 10 8 Substitute n = 3 into the recurrence relation. 3 = = + 9 Substitute 10 in place of. = Evaluate. = 1 11 Substitute n = 4 into the recurrence relation. 4 = = Substitute 1 in place of 3. = Evaluate. = Answer the question. The first five terms are 6, 8, 10, 1 and 14. Alternative method (short cut) 1 Begin with the number 6 and keep on adding until a set of numbers is obtained. Note: The relation n = n 1 +, 0 = 6, tells us that the first number in the set is 6 and every other number is obtained by adding to the previous number = = = = 14 1 Answer the question. The first five terms are 6, 8, 10, 1 and Maths Quest 11 Standard General Mathematics for the Casio ClassPad

23 WORKED EXAMPLE 14 Due to inflation, the price on a tub of yoghurt increases each year by c. In a certain year a tub of yoghurt cost 90c. a Set up a recurrence relation between consecutive years prices on yoghurt. b Find the price of yoghurt for each year for the net years. c Deduce an epression for the price for the nth year. THINK WRITE a 1 Write the general recurrence relation. a t n = t n 1 + d, t 0 = a Write the values of a and d. a = 90, d = 3 Substitute the known values of a and d into the recurrence relation. b 1 Substitute n = 1 into the recurrence relation and evaluate t 1. On the Sequence screen, tap ^. Complete the sequence as shown in the screen at right and tap #. Press E after each entry. t n = t n 1 +, t 0 = 90 b t n = t n 1 +, t 0 = 90 When n = 1 t 1 = t 0 + = 90 + = 9 Then tap #. 3 Write the net five terms of the recursive relation. 4 Answer the question, including the appropriate unit. t 1 = 9, t = 100, t 3 = 10, t 4 = 110, t = 11 The price of a tub of yoghurt for each of the net five years will be $0.9, $1.00, $1.0, $1.10 and $1.1. Chapter 93

24 c 1 Observe the terms from part b. Notice that each term is obtained by adding 90 to the product of a term s number and ; that is, t n = 90 + n. c t 1 = t = t 1 + = = 90 + t 3 = t + = = t 4 = t 3 + = = t = t 4 + = = 90 + t n = 90 + n Answer the question. The epression for the price, P, of a tub of yoghurt for the nth year is P = n If the linear recurrence relation t n = t n 1 + d, t 0 = a is rearranged to t n t n 1 = d, t 0 = a, the new relation is called a linear difference equation (since it shows the difference between two consecutive terms). Any linear difference equation has a corresponding linear equation of the form y = m + c, where y = t n, m = d and c = a. Therefore, y = m + c may be written as t n = dn + a. A linear difference equation t n t n 1 = d, t 0 = a has a corresponding linear equation: t = dn + a t n This correspondence is especially useful in solving problems describing real life situations. For instance, question c in the previous eample could have been solved using this formula in one step: t n = dn + a; d =, a = 90, so t n = n + 90 WORKED EXAMPLE 1 From the information shown in the following table, find: a a linear difference equation b a linear relation between t n and n. ebook plus s Tutorial int-0860 Worked eample 1 n t n THINK WRITE a 1 Find the difference between each pair of consecutive terms. a t 1 t = 4 t t = 4 t 3 t ( 1) = 4 t 4 t 3 9 ( ) = 4 Comment on the result obtained. The difference, d, is constant and equal to 4. 3 Write the linear difference equation. Difference equation: t n t n 1 = 4, t 0 = 7 b 1 Write the linear difference equation obtained in part a. Write the general form of the linear relation. 3 Substitute 4 in place of d and 7 in place of a. b t n t n 1 = 4, t 0 = 7 t n = dn + a t n = 4n Maths Quest 11 Standard General Mathematics for the Casio ClassPad

25 WORKED EXAMPLE 16 A computer purchased for $000 depreciates at $180 per year. a Find a difference equation which describes this situation. b Using a table of values, find out when the value of the computer will be less than 0% of its purchase price. THINK a 1 Write the general linear difference equation. Assign given values to a and d. Note: $000 corresponds to the original price, a. The value 180 corresponds to the rate of depreciation, d. The value is negative as the price decreases each year. 3 Substitute the known values into the linear difference equation. b 1 Rewrite difference equation as linear equation. Set up the table of values using a CAS calculator. First estimate the number of years you need. 3 On the Spreadsheet screen: Label column A: nvalue Enter the values of n into column A, beginning from 0 in increments of 1. Press E after each entry. Label column B: value In cell B complete the entry line as: = 180 A Then press E. Highlight cells B to B13 and tap: Edit Fill Range OK a b WRITE/DRAW t n t n 1 = d, t 0 = a t 0 = a = 000 d = 180 t n t n 1 = d, t 0 = a t n t n 1 = 180, t 0 = 000 t n = dn + a t n = 180n The depreciation rate is about $00 per year and the purchase price is $000; therefore we need roughly 000 = 10 years. Since the depreciation rate is 00 lower, we will determine the value for 11 years. Chapter 9

26 4 Copy results into the table of values. (Scroll down to see all entries in the table.) Determine what value 0% of the purchase price is. 6 Answer the question. Note: From the table, the value of the computer will be less than $400 ($380 to be eact) after 9 years from purchasing it. Year Value ($) Year Value ($) $000 = $ The computer will be less than 0% of its original value 9 years after purchasing it. REMEMBER 1. A linear recurrence relation describes a set of values, where each term after the first is obtained by adding the same number to the preceding term and is given by t n where t n = t n 1 + d, t 0 = a.. When the linear recurrence relation is rearranged to t n t n 1 = d, t 0 = a, it shows the difference between two consecutive terms and is called a linear difference equation. 3. Every linear difference equation has a corresponding linear equation t n = dn + a. EXERCISE D Linear recursion relationships 1 WE13 Write the first terms of each of the following recurrence relations: a t n = t n 1 + 6, t 0 = 1 b t n = t n 1 3, t 0 = 1 c t n = t n , t 0 = 1 d t n = t n 1 0.7, t 0 = e t n 1 = t n +.3, t 1 = 3 f u n = u n 1 + 4, u 0 = 0 g v n = v n 1 1, v 0 = 3 h n = n , 0 = 43 WE14 During the first 3 months of his life, a baby boy gains on average 0 g per week. a Set up a recurrence relation for the consecutive weekly weights of the boy who was kg when he was born. b Find the weight of the boy during his first 4 weeks of life. c Deduce the epression for the weight of the boy at the nth week from birth. ebookplus Digital doc SkillSHEET.3 Linear recursion relationships 96 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

27 3 Write the recursive relations for the information given in the following tables. i n ii n t n t n iii n iv n t n t n v n vi n t n t n WE1 For each of the relations in question 3, find: a the linear difference equation b the linear relation between t n and n. Find the first 4 terms for each of the following difference equations: a t n t n 1 = 3, t 0 = 1 b t n t n 1 =.69, t 0 =.19 c t n + 1 t n = 7, t 1 = 18 d t n + 1 t n = 4, t = 0 Questions 6 to 8 refer to the following recurrence relation: t n = t n 1 3, a =. 6 MC The first terms described by the above relation are: A 1, 4 B, 3 C, 1 D 1, E 3, 7 MC The values of d and a in the corresponding difference equation are: A, 3 B 3, C 3, D, 3 E 3, 8 MC Which of the following does not belong to the recursive relation that corresponds to the linear relation described above? A B C 1 D 4 E 7 9 WE16 A collection of coins originally valued at $0 appreciates each year by $3.70. a Find a difference equation which describes this situation. b How long will it take for the collection to double its original value? (Hint: Use a table of values to assist you.) 10 Suong is making a pyramid out of her tetas. The bottom row consists of 4 tetas and the net row consists of 1 tetas, and so on. a Find the recurrence relation for t n, the number of tetas in the nth row. b Find the epression for n, the number of tetas in any nth row. c Using the table of values, find in what row the number of tetas is: i a half ii a third iii a quarter of those in the first row. 11 Leo is involved in a weight-loss program where he is required to walk each day for a certain time. Upon joining the program, he has to walk for 18 minutes a day during the first week and etra minutes each consecutive week until he builds up his time to 4 minutes. Chapter 97

28 a Set up a recurrence relation defining this situation. b Create a table of values, which shows weeks and corresponding duration of the eercise (in minutes). Use the table to find: i the week when the initial duration of Leo s walk doubles ii the number of weeks taken to reach the target time. c Write the linear equation which relates the length of the eercise, t n, with the week number, n, and use it to verify your answers to part b. 1 Michelle and David are going to make scented candles for a Mother s Day stall. They have calculated that the setting-up cost will be $1 and then the cost of wa, wicks and perfume will be an additional $1.0 for each candle. They are planning to sell the candles at $3.0 each. a If the cost of making n candles is c n and the takings from selling n candles is t n, write the recurrence relations for both cost and takings. b Use a CAS calculator or a spreadsheet to find how many candles should be sold in order to break even (that is, for the set-up cost and cost of making n candles to equal the takings of selling n candles). c Write the difference equation for the profit. d Use a calculator or a spreadsheet to create a table of values showing the profit from selling up to 30 candles. E Simultaneous equations Consider the following problem. If two cassettes and three CDs cost a total of $4, what is the cost of one cassette and one CD? If we assume that the cost of each item is a whole number of dollars and that any price is possible, then each of these combinations could represent the solution to the problem. Of course, there could be many more answers if we also consider prices in dollars and cents. To be able to solve this problem, we need etra information in order to select the appropriate combination. For instance, if it is also known that four cassettes and one CD cost $34, then the only combination which will fit both descriptions is the situation where the cassette costs $6 and the CD costs $10. Cost of a cassette ($) Cost of a CD ($) We have just seen that it is impossible to solve one linear equation with two unknowns. There must be two equations with the same two unknowns in order for a solution to be found. Such equations are called simultaneous equations. Graphical solution of simultaneous equations If two straight lines intersect, the point of their intersection belongs to both lines and hence the coordinates of that point, and y, will represent the solution of two simultaneous equations which define the lines. 98 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

29 When we are solving simultaneous equations graphically, the accuracy of the solution is highly dependent on the quality of the graph. Therefore, all graphs must be drawn on graph paper as accurately as possible. It is good practice to verify any answer obtained from a graph by substituting it into the original equations, or to check using a CAS calculator. WORKED EXAMPLE 17 Solve the following pair of simultaneous equations: + y = 4 and y = 1 a graphically b graphically using a CAS calculator. THINK a 1 Rule up a set of aes. Label the origin and the and y aes. (See graph at step 7.) Find the -intercept for the equation + y = 4, by making y = 0. 3 Find the y-intercept for the equation + y = 4, by making = 0. Divide both sides of the equation by. 4 Plot the points on graph paper and join them with the straight line. Label the graph. (Refer to the graph at step 7.) Find the -intercept for the equation y = 1, by making y = 0. 6 Find the y-intercept for the equation y = 1, by making = 0. Multiply both sides of the equation by 1. 7 Plot the points on graph paper and join them with the straight line. Label the graph. a WRITE -intercept: when y = 0, + y = = 4 = 4 The -intercept is at (4, 0). y-intercept: when = 0, + y = y = 4 y = 4 y = The y-intercept is at (0, ). -intercept: when y = 0, y = 1 0 = 1 = 1 The -intercept is at (1, 0). y-intercept: when = 0, y = 1 0 y = 1 y = 1 y 1 = 1 1 y = 1 The y-intercept is at (0, 1). y y 4 (, 1) 1 y 1 4 Chapter 99

30 8 From the graph, read the coordinates of the point of intersection. 9 Verify the answer by substituting the point of intersection into the original equations. b 1 Rearrange each equation to make y the b subject. This can be done on the Main screen by completing the entry lines as: solve( + y = 4,y) solve( y = 1,y) Press E after each entry. The point of intersection between the two graphs is (, 1). Substitute = and y = 1 into + y = 4. LHS = + 1 RHS = 4 = + = 4 LHS = RHS Substitute = and y = 1 into y = 1 LHS = 1 RHS = 1 = 1 LHS = RHS In both cases LHS = RHS; therefore the solution set (, 1) is correct. On the Graph & Table screen, complete the entry line as: y1 = 4 y = 1 Tick the y1 and y boes and tap $. To find the point of intersection, tap: Analysis G-Solve Intersect 3 Write the solution. The point of intersection is (, 1). Therefore, = and y = 1. Parallel lines If two equations have the same gradient, they represent parallel lines. Such lines will never meet and so never have a point of intersection (that is, there is no solution). ebookplus s Digital doc SkillSHEET.4 Parallel lines 100 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

31 The following pair of equations, y = + 3 and y = + define two parallel lines; hence there is no solution. The graph to the right demonstrates that the straight lines never intersect with each other. Coincidental lines If two lines coincide, then there are an infinite number of solutions. For eample, consider the two straight lines given by the equations y = + 1 and 4 y =. Rearranging the second equation, we obtain the same line. 4 4 y = 4 y = 4 + y = y = y y y 1 4 y y 4 = y = + 1 Both equations when graphed will represent the same line they will coincide. Therefore, every point will represent the solution as there is not one unique point which will satisfy both equations. Algebraic solution of simultaneous equations When using algebra to solve simultaneous equations, the aim is to obtain one equation with one unknown from two equations with two unknowns by various algebraic manipulations. This can be done in two ways substitution and elimination as outlined below. Substitution method The method of substitution is easy to use when at least one of the equations represents one unknown in terms of the other. To solve simultaneous equations using the method of substitution: 1. Check that one of the equations is transposed so that one of the unknowns is epressed in terms of the other.. Substitute the transposed equation into the second equation. 3. Solve for the unknown variable 0 WORKED EXAMPLE 18 Use the method of substitution to solve the following pair of simultaneous equations: y = + 3 and 4 y =. THINK 1 Write the equations, one under the other, and number them. Substitute the epression ( + 3) from equation [1] for y into equation []. Note: By substituting one equation into the other, we are left with one equation and one unknown. 3 Solve for. (a) Epand the brackets on the LHS of the equation. WRITE y = + 3 [1] 4 y = [] Substituting ( + 3) into []: 4 ( + 3) = 4 3 = Chapter 101

32 (b) Simplify the LHS of the equation by collecting like terms. (c) Add 3 to both sides of the equation. 3 = = + 3 = 8 (d) Divide both sides of the equation by. 8 = = 4 4 Substitute 4 in place of into [1] to find the value of y. Substituting = 4 into [1]: y = Evaluate. = = 11 6 Answer the question. Solution: = 4, y = 11 or solution set (4, 11). 7 Verify the answer by subsituting the point of intersection into the original equations or use a graphics calculator. The answer was checked using a CAS calculator and found to be correct. If neither of the equations give one unknown in terms of the other, we can still use a method of substitution by first transposing one of the equations. Elimination method As the name suggests, the idea of the elimination method is to eliminate one of the variables. This is done in the following way. 1. Choose the variable you want to eliminate.. Make the coefficients of that variable equal in both equations. 3. Eliminate the variable by either adding or subtracting the two equations. Once this is done, the resulting equation will contain only one unknown which then can be easily found. WORKED EXAMPLE 19 Use the elimination method to solve the following: + 3y = 4 and 3y =. THINK WRITE 1 Write the equations, one under the other, and number them. Add equations [1] and [] in order to eliminate y. Note: y was eliminated since the coefficients of y in both equations were equal in magnitude and opposite in sign. [1] + []: + 3y = 4 [1] 3y = [] + 3y = 4 + ( 3y = ) 3 = 6 3 Divide both sides of the equation by = 3 3 = 4 Substitute the value of into equation []. Note: = may be substituted in either equation. Substituting = into []: 3y = 10 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

33 Solve for y. (a) Subtract from both sides of the equation. (b) Divide both sides of the equation by 3. 3y = 3y = 0 3y 0 = 3 3 y = 0 6 Answer the question. Solution: =, y = 0 or solution set (, 0). 7 Verify the answer by substituting the point of intersection into the original equations or using a graphics calculator. The answer was checked using a CAS calculator and found to be correct. Note: If there is no pair of equal coefficients, we can make them the same by multiplying or dividing one or both equations by an appropriate number. WORKED EXAMPLE 0 Find the values of and y for: + 3y = 4 and 3 + y = 10 a by elimination b using a CAS calculator. THINK WRITE a 1 Write the equations, one under the other, and number them. Decide which variable to eliminate, say. Eliminate. a + 3y = 4 [1] 3 + y = 10 [] (a) Multiply equation [1] by 3 and call the new equation [3]. (b) Multiply equation [] by and call the new equation [4]. [1] 3: 6 + 9y = 1 [3] [] : 6 + 4y = 0 [4] 3 Subtract equation [4] from equation [3]. [3] [4]: 6 + 9y = 1 (6 + 4y = 0) y = 8 4 Solve for y. Divide both sides of the equation by. Substitute the value of y into equation [1]. y 8 = 8 y = 8 Substitute y = = 4 6 Solve for. = 4 (a) Add to both sides of the equation. + = 4+ 4 or ( 1 3 ) into [1]: Chapter 103

34 (b) Simplify the RHS of the equation. = + (c) Divide both sides of the equation by. 0 = = 44 = = = or 4 ( ) 7 Answer the question. Solution: = 4, y = 1 3 or solution set 3 ( 4, 1 ). b 1 Write the equations. + 3y = y = 10 On the Main screen, using the soft keyboard, tap: ) {N Complete the entry line as shown at right and press E. 3 Write the solution. Solving + 3y = 4 and 3 + y = 10 for and y gives = and y = 8. Note: The answer can be verified by substituting the point of intersection into the original equations. REMEMBER Simultaneous linear equations can be solved either graphically or algebraically. 1. Graphical method Draw the straight lines representing the equations and find the coordinates of the point of intersection.. Algebraic methods (a) Substitution: Transpose one of the equations so that one of the unknowns is epressed in terms of the other and substitute into the second equation. (b) Elimination: Equate the coefficients of one unknown and eliminate it by either adding or subtracting the two equations. 104 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

35 EXERCISE E ebookplus Digital doc SkillSHEET.4 Parallel lines Simultaneous equations 1 WE17 Solve the following pairs of simultaneous equations: i graphically ii graphically using a CAS Calculator. a + y = 6 y = b 3 + y = 6 y = c + 3y = 3 + 3y = 1 d y = 6 4 y = 8 e = y + y = 0 f 3 + y = 6 y = 1 MC The pair of simultaneous equations y = and y = 1 will have: A 1 solution B solutions C no solutions D an infinite number of solutions E none of these 3 MC The pair of simultaneous equations y = 4 and y + = 0 will have: A 1 solution B solutions C no solutions D an infinite number of solutions E none of these 4 Complete the following statements. a If two lines with different gradients go through the origin, then the solution to the pair of simultaneous equations defining those lines would be (give coordinates). b If two lines have the same gradient but different y-intercept, then the pair of simultaneous equations defining such lines will have solution(s). c If both lines are defined by the equation y = m + c and have the same value of c but a different value of m, then the solution to such pairs of simultaneous equations will be (give coordinates). WE18 Solve the following pairs of simultaneous equations by the method of substitution. a y = y = 11 d = y 3 6y = 36 b = y 6 y = 10 e y 6 = 7 + 3y = 6 WE19 Use the method of elimination to solve each of the following: a + y = 3 4 y = 9 d 3 y = 1 3 6y = 9 g y = 0 4y = 9 b + y = 4y = e + 3y = 7 3y = 19 h + y = 8 + 7y = 3 c y = 3 6 y = 16 + f = 4 y y 3 = 13 c y = 7 y + = 1 f + 4y = y = 8 7 MC Nathan is solving a pair of simultaneous equations 3y = [1] and 3 + 4y = 10 [], using the elimination method. To eliminate one of the variables, he could multiply equation [1] and equation [] by: A and 3 respectively B 3 and 4 respectively C 3 and respectively D and 10 respectively E 4 and respectively 8 WE0 Solve each of the following pairs of equations using the elimination method. a 3y = y = d 1 +y y= 0 3y = 7 g y + 3 = 17 3y = 4 b y = y = 9 e y = y = h 1 3 y + y= y + = 0 c 3 1 y 3 + y= y = 3 f y = y = 4 Chapter 10

36 Question 9 refers to the diagram at right. 9 MC a The equation of line [1] is: 3 1 A y= C y= E y = + b The equation of line [] is: 3 B y = D y = A y = + B y = + 3 C y= D 3y = + 6 E 3y + = 6 c The point of intersection of the two lines has the coordinates: A 3 1 (, ) B, 1 y [] ( ) C (, 1) D ( 3, 1 ) E (, 3 ) 10 The lines y + 8 = 0 and 1 = 0 intersect at the point: A ( 1, 8) B ( 8, 1) C (0, 0) D (8, 1) E (1, 8) 11 For the pair of simultaneous equations 3y = 7 and 3 = y the solution is: A =, y = 1 B = 1, y = 3 C = 1, y = D =, y = 3 E =, y = 1 3 [VCAA 004] [VCAA 003] 1 Two lines have equations y = and y = + respectively. The point that lies on both of these lines is: A ( 10, ) B (, ) C (0, ) D (, ) E (10, ) 13 Here are 3 simultaneous equations with 3 unknowns. Find the values of, y and z. + 3y z = y + z = 4y + z = 1 [VCAA 00] 14 The graph at right represents the following information. At a certain factory, the cost of producing n 1 pairs of adult s shoes C is given by C 1 = n 1 and the cost of producing n pairs A of children s shoes is given by C = 13 + n. a Which of the lines, A or Z, represents the cost of 198 Z producing adult s shoes and which one represents the 13 cost of producing children s shoes? b Why do you think the lines representing cost do not start 0 n No. of pairs of shoes at the origin and what could numbers 13 and 198 on the vertical ais represent? c Find the number of shoes that should be produced so that the cost of production of both types is the same. d Find the cost for producing the number of pairs in part c. e Where on the graph can this number, as in part c, and this cost, as in part d, be found? f What does the portion of the graph to the left of the point of intersection tell us? g What does the portion of the graph to the right of the point of intersection tell us? h What could the coefficients of n in the equations of the cost mean, in terms of production? Cost ($) [1] Maths Quest 11 Standard General Mathematics for the Casio ClassPad

37 F Solving problems using simultaneous equations Simultaneous equations are used to solve a variety of problems containing more than one unknown. Here is a simple algorithm which can be applied to any of them: 1. Identify the variables.. Set up simultaneous equations by transforming written information into algebraic sentences. 3. Solve the equations by using the substitution, elimination or graphical methods. 4. Interpret your answer by referring back to the original problem. WORKED EXAMPLE 1 Find two consecutive numbers which add up to 99. THINK WRITE 1 Define the two variables. Let = the first number. Let y = the second number. Formulate two equations from the information given and number them. Note: Consecutive numbers follow one another and differ by 1. Hence, if is the first number, the net number will be + 1 that is, y = Substitute the epression ( + 1) from equation [] for y into equation [1]. + y = 99 [1] y = + 1 [] Substituting ( + 1) into [1]: = 99 4 Solve for. (a) Simplify the LHS of the equation by collecting like terms. (b) Subtract 1 from both sides of the equation. (c) Divide both sides of the equation by. Substitute 49 in place of into equation [1] to find the value of y. + 1 = = 99 1 = = = 49 Substituting = 49 into equation []: y = Evaluate. y = = 0 7 Verify the answer by checking that the two values are consecutive and that they sum and 0 are consecutive numbers = 99 The obtained values satisfy the problem. 8 Answer the question. The two consecutive numbers which add up to 99 are 49 and 0. Chapter 107

38 WORKED EXAMPLE Two hamburgers and a packet of chips cost $8.0, while 1 hamburger and packets of chips cost $.90. Find the cost of a packet of chips and a hamburger. THINK WRITE 1 Define the two variables. Let = the cost of one hamburger. Let y = the cost of a packet of chips. Formulate an equation from the first sentence and call it [1]. Note: 1 hamburger costs $, hamburgers cost $. Thus, the total cost of cost of hamburgers and 1 packet of chips is + y and it is equal to $ Formulate an equation from the second sentence and call it []. Note: 1 packet of chips costs $y, packets cost $y. Thus, the total cost of packets of chips and 1 hamburger is + y and it is equal to $ Eliminate the variable. (a) Multiply equation [] by and call it equation [3]. ebook plus s Tutorial int-0861 Worked eample + y = 8.0 [1] + y =.90 [] [] : ( + y =.90) + 4y = [3] (b) Subtract equation [1] from equation [3]. [3] [1]: + 4y = ( + y = 8.0) 3y = 3.60 (c) Divide both sides of the equation by 3. Solve for. 3y 360. = 3 3 y = 1.0 (a) Substitute y = 1.0 into equation [1]. Substituting 1.0 into [1]: = 8.0 (b) Subtract 1.0 from both sides of the equation. (c) Divide both sides of the equation by. 6 Answer the question and include appropriate units. 7 Verify the answer by substituting the values into the original equations or using a graphics calculator = = = = 3.0 A hamburger costs $3.0 and a packet of chips costs $1.0. The answer was checked using a calculator and found to be correct. It is etremely important to be consistent with the use of units while setting up equations. For eample, if the cost of each item is epressed in cents, then the total cost must also be epressed in cents. 108 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

39 REMEMBER To solve problems involving simultaneous equations, follow these steps: 1. Identify and define the variables.. Transform written information into algebraic statements. 3. Solve the pair of equations graphically or algebraically using the methods of substitution or elimination. 4. Interpret the result by relating the answer back to the problem.. Always make sure the numbers in the equations are in the same units. EXERCISE F ebook plus s Digital doc Spreadsheet 11 Simultaneous equations Solving problems using simultaneous equations 1 WE1 Find two consecutive numbers which add up to 89. The sum of two numbers is 0, while their difference is 1 of their sum. Find the numbers. 3 One number is twice as large as the other. Five times the smaller number is 4 more than the larger number. Find the numbers. 4 The sum of two consecutive even numbers is the same as the product of two consecutive numbers, of which 7 is the largest. Find the numbers. When three times the first number is added to twice the second number, the result is 13. Four times the difference of those numbers is 44. Find the numbers. 6 Half of the sum of two numbers is 6 less than the first number. One-third 1 of their difference 3 is one less than the second number. Find the numbers. 7 Five times the first number is twice as large as 4 times the second number. When the difference of the two numbers is multiplied by 0, the result is 3. Find the numbers. 8 The average of two numbers is 11. Their difference is 3 more than that. Find the numbers. 9 A rectangle s length is cm more than its width. If the perimeter of a rectangle is 4 cm, find its dimensions and, hence, its area. 10 An isosceles triangle, with equal sides each cm longer than the third side, is constructed on a side of a square. If the perimeter of the triangle is 8 cm, find the perimeter of the square. 11 In the rectangle at right, find the values of and y. Hence, determine the perimeter. ( ) cm ( 1) cm 10 cm ( y) cm 1 The sides of an equilateral triangle have the following lengths: ( + y) cm, ( 3) cm and (3y 1) cm. Find the perimeter of the triangle. Chapter 109

40 13 The perimeter of a rhombus ABCD is 10 cm longer than the perimeter of an isosceles triangle ABC. Find the length of AC, the diagonal of a rhombus, if it is cm smaller than its side. B A C D 14 A table consists of columns and rows. Each of its cells is a rectangle with length cm and width y cm. The perimeter of the table is 70 cm and the total length of interior lines is 6 cm. a Draw a diagram to represent the above information. b Find the dimensions of each cell and comment on its shape. 1 Phuong conducts a survey. She asks 7 people whether or not they use the Internet at home. There were three times as many people who answered Yes as those who answered No. Find the number of people in each category and hence help Phuong to complete the following statement: According to the survey (insert fraction) of the population uses the Internet at home. 16 WE At the end of the day, two shop assistants compare their sales. One sold toasters and sandwich-makers for a total of $149.6, while the other sold 3 of each for a total value of $ Find the price of each item. 17 At lunch time, Michael bought egg and bacon rolls and egg and vegetable rolls. He received $3.0 change from a $0 note. If an egg and bacon roll costs 70c more than an egg and vegetable roll, how much did each roll cost? 18 In an aquatic centre, a pool and spa entry is $3.0, while pool, spa, sauna and steam room entry is $.0. At the end of the day, a cashier finds that she sold 193 tickets altogether and her takings are 40c short of $800. How many of each type of ticket were sold? 19 Spiro empties his piggy bank. He has 4 coins, some of which are c coins and some of which are 10c coins, to the total value of $.0. How many c coins and how many 10c coins does he have? 110 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

41 0 Maya and Rose are buying meat for a picnic. Maya s family likes lamb more than pork, so she buys 3 kg of lamb and only half as much pork. Rose s family have different tastes, so she buys 4. kg of pork and one-third as much lamb. If Maya spends $13.0, which is $8. less than Rose spends, what is the cost of 1 kg of each type of meat? 1 Rachel is 4 times as old as her brother Nathan. If she will be twice as old as him in two years time, how old is she now? Bella and Boris are celebrating their th wedding anniversary. Today, their combined age is eactly 100. If Boris is 4 years older than Bella, how old was his bride on the day of their wedding? 3 Interpreting Pty Ltd translates each English tet into both French and Japanese. It takes a French interpreter 0.6 hours to translate a page of any scientific tet and 1 hour to translate a page of fiction. A Japanese interpreter needs 0.9 hours to translate scientific tet and 1. hours for fiction. If the French interpreter works 8 hours a day, while the Japanese interpreter is prepared to take some of her work home and spend up to 1. hours per day altogether, what is the maimum number of pages of each type of tet that can be translated each day by Interpreting Pty Ltd? 4 Sasha is making dim sims and spring rolls for his guests. He is going to prepare everything first and then cook. On average it takes 0. hours to prepare one portion of dim sims and 0. hours to prepare one portion of spring rolls. He needs 0.0 hours and 0.1 hours to cook each portion of dim sims and spring rolls respectively. If he spends hours on preparation and 1 minutes on cooking, how many portions of dim sims and spring rolls does Sasha make? As a space shuttle or rocket is launched, the astronauts inside eperience a great deal of stress. An astronaut s heart pumps twice as much blood per minute during rocket launch as under normal conditions. That is, the heart pumps about 4.7 litres more blood each minute during a launch than when the astronaut is at rest on the ground. a Write two equations linking the heart s rate of pumping blood under normal conditions to that during launch. b Solve these equations to find the heart s rate of pumping blood for the astronaut under both normal and launch conditions. Chapter 111

42 SUMMARY A linear equation is an equation which contains a pronumeral raised to the power of 1. are solved by using inverse operations. When solving linear equations, the order of operations process, BODMAS, is reversed. Solving problems using linear equations To solve worded problems using linear equations, follow these steps: 1. Identify the variables.. Set up an equation by transforming the written information into an algebraic statement or statements. 3. Solve the equation. 4. Interpret the result by relating the answer back to the original problem. Substitution and transposition in linear relations If all the variables but one are known in a formula, the value of the unknown variable may be found by substituting the known values into the formula. Transposition is the rearrangement of the formula. The subject of the formula is the variable which is by itself on one side of the equation, while all other variables are on the other side of the equation. Linear recursion relationships A linear recurrence relation describes a set of values, where each term after the first term is obtained by adding the same number to the preceding term. It is given by the relation t n = t n 1 + d, t 0 = a. When the linear recurrence relation is rearranged to t n t n 1 = d, t 0 = a, the new relation shows the difference between two consecutive terms and is called a linear difference equation. Every linear difference equation has a corresponding linear equation t n = dn + a. Tables of values, generated by a CAS calculator or a spreadsheet, are valuable tools in solving problems involving linear recursion. Simultaneous equations Simultaneous linear equations can be solved either graphically or algebraically. 1. Graphical method Draw the straight lines representing the equations and find the coordinates of the point of intersection.. Algebraic methods (a) Substitution: Transpose one of the equations so that one of the unknowns is epressed in terms of the other and substitute into the second equation. (b) Elimination: Equate the coefficients of one unknown and eliminate it by either adding or subtracting the two equations. Solving problems using simultaneous equations 1. Identify and define the variables.. Transform written information into algebraic statements. 3. Solve the pair of equations graphically or algebraically, using the methods of substitution or elimination. 4. Interpret the result by relating the answer back to the problem.. Always make sure the numbers in the equations are in the same units. 11 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

43 CHAPTER REVIEW MULTIPLE CHOICE 1 The solution to the equation = 1is: 3 A 1 B C 3 D E 6 To solve 1 3 =6, the following operations 4 could be performed to both sides of the equation: A Add 1; multiply by 4; divide by 3 B Multiply by 4; divide by 3; subtract 1 C Multiply by 4; divide by 3; subtract 1 D Subtract 1; multiply by 4; divide by 3 E Multiply by 4; subtract 1; divide by 3 3 The equation which is the same as (3 1) = + 3 is: A 6 = + 1 B 11 = C = 3 D = E 11 = 3 4 The number that satisfies the equation 1 + = is: 3 4 A 4 B 3 C 3 D 4 E The number that satisfies the equation + 1 = is: A 1 B 6 C D E 6 6 If 3 times a number subtracted from 6 gives 9, then the number is: A B 1 C 1 D 1 E The perimeter of a regular heagon is 1.6 cm more than the perimeter of a square with the same side length. The length of the side of a heagon is: A.1 cm B 3.1 cm C 1.6 cm D 1.6 cm E 6.3 cm 8 When half a number is subtracted from 8, the result is the same as adding double that number to. The equation that matches this information is: A 8 8= + B 8 = + C 8 = + D +8= 8 + E 8= If A = B + 3 is transposed to make B the subject, then: 4 3 A B= A B B = 4 A + 3 C B = A 3 D B= 4A E B= 4A+ 3 3 Questions 10 and 11 refer to the shape below. 10 Using π =, the perimeter of a 7 certain shape is given by 11 P = When transposed to make the subject, is: A 7 P B 3 C 7P P D 7 P E 7 ( P 3) If the perimeter of the above shape is 8 cm, then is equal to: A 4 cm B 4 cm 7 C 4 cm D 3 11 cm E 1.7 cm 1 The linear equation corresponding to the difference equation t n t n 1 = 6, t 0 = 3 is: A t n = 6n 3 B t n = t n 1 + 6, t 0 = 3 C t n = 6 3n D t n = t n 1 3, t 0 = 6 E t n = 6t t n 1 = 3 13 The fourth term of the recurrence relation t n = t n 1, t 0 = 1 is: A B 0 C 8 D 6 E 4 14 A second-hand car purchased for $000 loses $00 of its value each year. This situation is not described by: A t n = t n 1 00, t 0 = 000 B t n t n 1 = 00, t 0 = 000 C t n = n D t n = 00n E t n + 1 = t n 00, t 1 = A difference equation is defined by f n + 1 f n =, where f 1 = 1 The sequence f 1, f, f 3, is: A, 4, 3 B 4, 9, 14 C 1, 6, 11 D 1, 4, 9 E 1, 6, 11 EXAM TIP 46% of all students answered this question correctly. [Assessment report 006] [VCAA 006] Chapter 113

44 16 The sum of solutions of the pair of simultaneous equations y + = 1 and y = 6 is: A 36 B 1 C 0 D 4 E If y = 3 4 and y = + 4, then the values of and y respectively are: A 9, 31 B 9, 31 C 31, 9 D 9, 31 E 9, The point of intersection of the lines is: A (1, 3) y 1 1 ( ) 1 1, B 1, 3 ( ) 1 1 (, 9 9 ) C 1 3 D 1 3 E (, 3) 19 The statement below which is not true for the pair of simultaneous equations y + = and 3 y = 6 is: A The sum of the numbers is. B 3 times the first number is 6 larger than the second number. C 3 times 1 number is 6 smaller than the other number. D The difference between 3 times 1 number and the other is 6. E When 1 number is subtracted from, the other number is obtained. 0 Which one of the following pairs of simultaneous linear equations has no solution? A 3 y = 4 + y = 9 C + 3y = 0 y = 7 B y = 1 4 y = 3 D 3y = y = 8 E 4 + y = 6 y = 0 [VCAA 007] 1 The point of intersection of two lines is (, ). One of these two lines could be: A y = 0 B + y = 8 C + y = 0 D y = 4 E y = 0 [VCAA 006] The sum of two numbers is 4 and their difference is 4. The smaller of the numbers is: A 3 B 17 C 18 D 19 E 4 3 Ben is 1 year short of being twice as old as Ester. If their ages total 0 years, Ben is: A 11 B 1 C 13 D 14 E One afternoon at the beach Mr Smith bought four ice creams and three drinks for his family at a cost of $1.40. Mrs Brown bought five of the same ice creams and two of the same drinks for $0.80. Based on these prices, the cost of one drink is: A $.80 B $.90 C $3.00 D $3.30 E $3.40 [VCAA 00] SHORT ANSWER 1 Solve for = Before opening the store, a cashier makes sure that his register contains at least $ in change. He counts a number of 10c coins, twice as many c coins and 4 times as many 0c coins to the total value of eactly $. How many coins of each type does he count? 3 A building company charges a $300 set fee plus $00 a day while it is working on a project within the time limits that are specified by a contract. If the project is completed earlier than the set time, the company will still charge $00 for each of the remaining days. However, if the project is not completed by the due date, the company will pay a $13 penalty for each etra day until the work is done. From the given information, construct a set of formulas for the total cost of work, T,, the number of days it takes to complete the job according to the contract, n, and the number of etra days, e. 4 Transpose each of the following formulas to make the pronumeral, indicated in brackets, the subject. (If two pronumerals are indicated, make a separate transposition for each.) a 6 1y + 1 = 0 () b 7 (3 4 d) ) 8 (e e+ 7) = 3 (d, e) 3 The linear recurrence relation is given by t n = t n 1 + 3, t 0 = 7. a Write the first 4 terms of the relation. b Write the corresponding difference equation. c Write the rule connecting the nth term with n. Hence, find the 0th term. 114 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

45 6 Anna has invested $000 for a 4.% simple interest that is credited to her account at the end of each financial year. a Write the difference equation to represent this situation. b Anna is planning to visit her Aunt Rose who lives in Perth. She estimates that she will need $00. If Anna is going to use her investment to finance the trip, how soon will she be able to go to Perth? 7 y a Find the equations of the two lines shown on the diagram. b Find the coordinates of the point of intersection (the diagram is not drawn to scale). 8 Solve the following simultaneous equations. a 6 + y = 1 y = b 8y 4 = y = c 1 3 3y = 30 + y = 4 9 Jessica is 3 years older than Rebecca. In years she will be 3 times as old as Rebecca was years ago. Find the girls present ages. EXTENDED RESPONSE 1 Adrian has begun a new job as a car salesman. His fortnightly wage is calculated in two parts: the first, a set amount of $600; the second, % of sales made each fortnight. a Write the rule describing Adrian s fortnightly wage. b How much can Adrian epect to take home if he makes: i $0 000 ii $6 000 iii $ in sales in a particular fortnight? c How much must Adrian make in sales in order to obtain a fortnightly wage of: i $1300 ii $1800 iii $400? Brett, also a salesman in the motor vehicle industry, is paid a fortnightly salary of $860 regardless of sales made. d Compare Adrian s fortnightly wage to Brett s fortnightly salary. e Write the rule describing Brett s fortnightly salary. f How much would Adrian have to make in sales in one fortnight to obtain the same amount as Brett takes home? Joseph has $1 000 to invest. He does not want to keep all of his eggs in the one basket, so he decides to split the money in the following ways. He puts some of his money in the bank, which offers an interest rate of 6% p.a., and the remainder into a building society, which offers an interest rate of 11% p.a. If Joseph plans to take a trip to Queensland, costing $100, and he wants to pay for the trip using only the interest earned from his investments after 1 year, how must he split his $1 000? 3 Michael wishes to rent a car for a long weekend. The cost, C, of renting a Toyota Corolla from company A is given by C = n, and the cost of renting from company B C = n, where n is the number of kilometres travelled. a Which company, A or B, does line [1] represent? b What could the numbers and 40 represent? c What does the point of intersection of lines [1] and [] represent? d Find the coordinates of the point of intersection. e If Michael decides to travel along the Great Ocean Road, which is about 30 km each way, from which company, A or B, should he rent so that he pays less? f Net long weekend, Michael is planning to go to Phillip Island, which is about 10 km each way. From which company should he rent this time? Cost ($) C 40 0 Number of km [] [1] n Chapter 11

46 g Eplain to Michael how he can decide from which company to rent, if he knows the approimate distance he intends to travel, without doing any calculations. h Write the formula for d, the difference between the cost of renting the car from the two companies (A or B). i Write the difference equation which corresponds to the equation in part h. j Use the difference equation to generate a table of values for distances from 0 to 1000 km inclusive, with increments of 100 km. Hence, find the distance for which the cost of renting from company A will eceed the cost of renting from company B by more than 10 km. 4 A clothing manufacturer finds that the cost, C dollars, of producing shirts is given by the equation C = a Determine the cost of producing 400 shirts. b Determine the maimum number of shirts that can be EXAM TIP Generally well answered produced for $3000. [Assessment report 004] c Assuming all the shirts are sold, the revenue, R dollars, from the sale of shirts produced is given by an equation R = 3. A graph of the revenue equation R = 3 for is drawn on the aes below. On these same aes draw a graph of the cost equation C = for d Determine the number of shirts that need to be produced and sold for the manufacturer to break even. EXAM TIP the graph. This could have been calculated or read off [Assessment report 004] e Given the cost equation is C = and the revenue equation is R = 3, write an equation for the profit, P dollars, from the production and sale of shirts. EXAM TIP The most common error was in simplifying P = R C where brackets were not applied to the cost function. This gave the incorrect answer of P = This was awarded one of the two available marks. The relationship of P = R C does not seem to be well understood by a number of students. [Assessment report 004] Dollars ($) f Calculate the profit from the production and sale of 34 shirts. g The manufacturer also produces jackets. They receive an order for 0 jackets. The cost of producing the 0 jackets is $4800. Determine the selling price per jacket to achieve an overall profit of $3000. [VCAA 004] Number of shirts R 3 EXAM TIP This mark was awarded for a correct substitution in the student s equation for part e. [Assessment report 004] EXAM TIP Several students wrote this as $31. which was awarded the mark, although teachers should address such notation in class. [Assessment report 004] 116 Maths Quest 11 Standard General Mathematics for the Casio ClassPad

47 Novak Novelties manufactures a variety of children s 3-D puzzles. The director of the company has asked his assistants Caitlin, Bridget and Emese to prepare a report on production costs, epenses and returns on the puzzles. Each puzzle costs the company $1 to produce. In addition, the company has monthly overheads of $ The selling price of each puzzle is $4. a Write an equation describing the epenses; that is, the total cost, C, of producing n puzzles each month. b Write an equation describing the selling price of n puzzles. c Plot and label the graph of the equation obtained in part a. Does it commence at the origin? Eplain. d Plot and label the graph of the equation obtained in part b on the same ais. Does it commence at the origin? Eplain. e The point of intersection of the two lines on your graph is called the break-even point. Eplain what this means in terms of the given problem. f Find the coordinates of the break-even point (point of intersection). g Shade the portion between the two lines to the left of the break-even point. Eplain what this portion represents. h Shade the portion between the two lines to the right of the break-even point. Eplain what this portion represents. Profit may be defined as the selling price minus the total cost. i Write an equation describing the profit obtained, P, after selling n puzzles. j Determine whether a profit or loss is made when: i 400 ii 600 iii 800 iv 1000 puzzles are sold in a particular month. ebook plus s Digital doc Test Yourself Chapter Chapter 117

48 ebookplus ACTIVITIES Chapter opener Digital doc 10 Quick Questions: Warm up with ten quick questions on linear equations. (page 71) A Tutorial WE int-087: Watch how to solve a linear equation involving fractions. (page 7) Digital docs SkillSHEET.1: Practise solving linear equations. (page 76) SkillSHEET.: Practise finding the lowest common multiple. (page 77) B Solving problems using linear equations Tutorial WE8 int-088: Watch how to define variables, set up linear equations and solve them to answer an application question. (page 80) Digital doc WorkSHEET.1: Construct and solve linear equations algebraically. (page 83) C Substitution and transposition in linear relations Tutorial WE 11 int-089: Watch how to graph the relationship between interest and rate of interest and use the graph to determine amounts of interest for set rates and rates for set interest amounts. (page 86) Digital doc WorkSHEET.: Solve and transpose linear equations. (page 91) D Linear recursion relationships Interactivity Linear recursion relationships int-0803: Consolidate your understanding of linear recursion relationships. (page 9) Tutorial WE 1 int-0860: Watch how to use the information provided in a table determine the linear difference equation and the linear relation between the term and the number of terms. (page 94) Digital doc SkillSHEET.3: Practise linear recursion relationships. (page 96) E Simultaneous equations Digital doc SkillSHEET.4: Practise working with parallel lines. (page 100, 10) F Solving problems using simultaneous equations Tutorial WE int-0861: Watch how to use the elimination method to solve a set of simultaneous equations. (page 108) Digital doc Spreadsheet 11: Investigate solving simultaneous equations. (page 109) Chapter review Digital doc Test Yourself Chapter : Take the end-of-chapter test to test your progress. (page 117) To access ebookplus activities, log on to Maths Quest 11 Standard General Mathematics for the Casio ClassPad