SIMULTANEOUS LINEAR EQUATIONS A. Construction of Formulae to Describe a Linear Relationship Two examples can be given as an introduction Ð one showing formula of the type y = mx , passing through the origin, the other of the type y = mx + c, crossing the the y axis at (0,c). On board: Example 1. Tomatoes sold in 2 kg packets. Compare no. of bags sold with their weight. (a) Copy and complete a table. Number of packets (N) Weight of tomatoes (W) (b) (c) (d) 1 2 Write a formula. Find the weight of 20 packets Draw straight line graph through O. 2 3 4 5 6 (W = 2N) (40 kg) (Watch scales and naming axes) Example 2. Hiring a chain saw. Basic charge £5, plus £3 per day (a) Copy and complete a table. Number of Days (D) Charge (C) (b) (c) (d) 1 8 2 3 11 4 5 6 Write a formula. (C = 3D + 5) Find the charge for 10 days. (£35) Plot points from table and draw straight line graph. Extend line to pass through (0,5) (Watch scales and naming axes) The significance of the Ò5Ó in C = 3D + 5 should be pointed out. Other lines such as C = 2D + 4, C = 6D Ð 1 etc. should be considered and students asked where they would cross the vertical (D) axis. Exercise 1 may now be attempted. Mathematics Support Materials: Mathematics 2 (Int 2) - Staff Notes 9 B. Solving Simultaneous Equations in Two Variables Graphically Introduction : Drawing Straight Line Graphs (Revision) ¥ choose three points which fit the equation of the line ¥ plot the points on squared paper ¥ draw a straight line through them Example 1. y = 4x ¥ Choose say, x = 0, x = 2 and x = 4 to obtain y = 0, y = 8 and y = 16 ¥ Plot (0,0) (2,8) (4,16) ¥ Join the points and extend line into negatives (Watch scales and naming axes) Example 2. y = 2x + 1 ¥ Choose say, x = 0, x = 3 and x = 5 to obtain y = 1, y = 7 and y = 11 ¥ Plot (0,1) (3,7) (5,11) ¥ Join the points and extend line (Watch scales and naming axes) Example 3. 2x Ð y = 5 Not quite as easy ¥ Choose say, x = 0, x = 3 and x = 5 * solve 3 simple equations to obtain y = Ð5, y = 1 and y = 5 ¥ Plot (0,Ð5) (3,1) (5,5) ¥ Join the points and extend line (Watch scales and naming axes) Exercise 2 may now be attempted. Finding the Point of Intersection of 2 Straight Lines Draw graphs of these equations to solve the pairs of simultaneous equations: Example 1. x+y=3 y=x+1 As in Exercise 2 Ð Choose suitable points for x + y = 3 and draw the line on a coordinate diagram on squared paper. Now, choose suitable points for y = x + 1 and draw the line on the same coordinate diagram. Both lines are seen to cross at the point (1,2) Example 2. x+y=2 3x Ð 2y = 11 Choose suitable points for x + y = 2 and draw the line on a coordinate diagram on squared paper. Now, carefully choose suitable points for 3x Ð 2y = 11 and draw the line on the same coordinate diagram. Both lines are seen to cross at the point (3,Ð1) Exercise 3 may now be attempted. Mathematics Support Materials: Mathematics 2 (Int 2) - Staff Notes 10 C. The Significance of the Point of Intersection of 2 Graphs A straightforward example, such as the one below, should be given as an introduction . Example: 2 ice creams and 1 ice lolly cost £1á30 5 ice creams and 3 ice lollies cost £3á40 Students should be encouraged to construct formulae to represent the relationships. i.e. 2x + y = 1á30 and 5x + 3y = 3á40, where x is the cost of 1 ice cream and y is the cost of 1 ice lolly. The graphs can then be drawn and the significance of the point of intersection can be stressed. A second example. as shown on the next page, could be used. Prices on the Dunoon Car Ferry: 3 cars and 1 motor cycle cost £35 for a single crossing 2 cars and 3 motor cycles cost £35 for a single crossing What is the cost for my own car ? The following example could be used to introduce exercise 4A Example Mr. Adam and Mrs. Bryce each own a locksmithÕs shop. For emergency repairs, Mr. Adam charges £15 per hour plus a callÐout fee of £10. For the same repairs, Mrs. Bryce charges £10 per hour plus a callÐout fee of £20. (a) Make two tables to show the prices for up to a 5 hour callÐout at AdamÕs and BryceÕs. Adam 0 1 2 3 4 5 10 25 40 55 70 85 Bryce 0 1 2 3 4 5 20 30 40 50 60 70 (b) Draw the straight line graph for both locksmith companies on the same coordinate diagram. (c) For how many hours callÐout is the cost the same at both shops ? (2 hrs) (d) If you needed a callÐout for a job which you knew would take a long time to complete, which shop would you phone to in order to save money ? (Mrs. Bryce) Exercise 4A may now be attempted. Mathematics Support Materials: Mathematics 2 (Int 2) - Staff Notes 11 The following example could be used to introduce exercise 4B Example (a) 2 C.D.Õs and 1 cassette cost £45. 1 C.D. and 4 cassettes cost £40. Let the cost of a C.D. be £x and the cost of a cassette be £y. Write down two equations in terms of x and y. (2x + y = 45; x + 4y = 40) (b) Draw the two straight lines which the equations represent on the same coordinate diagram using suitable points on each line. (c) Use your graph to find the cost of a C.D. and the cost of a cassette. (£20) (£5) (20,5) Exercise 4B may now be attempted. Mathematics Support Materials: Mathematics 2 (Int 2) - Staff Notes 12 D. Solving Simultaneous Linear Equations Algebraically Eliminating x or y by Addition, or by first Multiplying both sides of o n e Equation by Ð1. Example 1. Solve: x + y = 14 x Ðy= Here is the ideal situation .... +y and Ðy. Simply ADD both equations to eliminate y. x + y = 14 xÐy=8 1 + 2 => Look for the coefficient of one term being the negative of another. Ð Ð 1 2 2x = 22 x = 11 Substitute x = 11 into either equation 1 or 2. Giving 11 + y = 14 y=3 e.g in 1 Point of intersection (11,3) x + 3y = 10 x Ð 3y = 4 Again, the ideal situation .... +3y and Ð3y. Simply ADD both equations to eliminate y. x + 3y = 10 Ð x Ð 3y = 4 Ð CHECK by substitution! Example 2. Solve: Look for the coefficient of one term being the negative of another. 1 2 2x = 14 x=7 Substitute x = 7 into either equation 1 or 2. Giving 7 + 3y = 10 3y = 3 y=1 e.g in 1 Point of intersection (7,1) x + 2y = 4 Ð 1 x Ð 3y = Ð1 Ð 2 NOT quite the ideal situation .... +2y and Ð3y not good. Multiply BOTH SIDES of equation 2 by Ð1. x + 2y = 4 Ð 1 2 becomes...... Ðx + 3y = 1 Ð 3 CHECK by substitution! Example 3. Solve: Mathematics Support Materials: Mathematics 2 (Int 2) - Staff Notes Look for the coefficient of one term being the negative of another. 13 Now, the ideal situation .....an x and a Ðx ADD both equations to eliminate x. x + 2y = 4 Ð 1 Ðx + 3y = 1 Ð 3 5y = 5 y=1 Substitute y = 1 into either equation 1 or 2. e.g in 1 Giving x+2=4 x=2 Point of intersection (2,1) CHECK by substitution! Exercise 5A may now be attempted. Eliminating x or y by first Multiplying both sides of one Equation by a Suitable Number, then Adding. Example 1. Solve: x + 2y = 8 Ð 1 3x Ð y = 17 Ð 2 NOT quite the ideal situation .... +2y and Ðy not good. Multiply BOTH SIDES of equation 2 by 2. x + 2y = 8 Ð 1 2 becomes...... 6x Ð 2y = 34 Ð 3 Now, the ideal situation .....a 2y and a Ð2y ADD both equations to eliminate y. x + 2y = 8 Ð 1 6x Ð 2y = 34 Ð 3 7x = 42 x=6 Substitute x = 6 into either equation 1, 2 or 3. Giving 6 + 2y = 8 2y = 2 y=1 e.g in Point of intersection (6,1) Example 2. Solve: 3x + 2y = 13 Ð x+y=5 Ð Look for the coefficient of one term being the negative of another. 1 CHECK by substitution! 1 2 This can be best be solved by either :Ð (i) multiplying 2 by Ð3 or (ii) multiplying 2 by Ð2 Answer (3,2) Exercise 5B may now be attempted. Mathematics Support Materials: Mathematics 2 (Int 2) - Staff Notes 14 Eliminating x or y by first Multiplying both sides of one or both Equations by a Suitable Number, then Adding. Example . Solve: 2x Ð 3y = 7 Ð 1 3x Ð 2y = 13 Ð 2 Point out that in a situation like this, it is not enough to change one of the equations. BOTH equations have to be be multiplied ! In this case, suggestions should be sought Ð e.g. multiply 1 by 2 and 2 by Ð3. 1 becomes....... 4x Ð 6y = 14 Ð 3 2 becomes...... Ð9x + 6y = Ð39 Ð 4 ADD both equations to eliminate y. Ð5x = Ð25 x=5 Substitute x = 5 into either equation 1, 2, 3 or 4. Giving 10 Ð 3y = 7 Ð3y = Ð3 y=1 e.g in Point of intersection (5,1) 1 CHECK by substitution ! Exercise 5C may now be attempted. Mathematics Support Materials: Mathematics 2 (Int 2) - Staff Notes 15 E. Using Simultaneous Equations to Solve Problems This part is a continuation of the solving of simultaneous equations in Exs. 5A, 5B and 5C. Now though, a situation (or a picture) is given, from which both equations have to be derived before being solved. Some problems may be simplified by drawing Ð Total Cost = £10á30 i.e. Total Cost = £15á20 3p + 2b = 10á30 2p + 3b = 15á20 Solve as before. (1 pencil = 10 pence 1 = book £5) From this, the cost of 20 books and 14 pencils etc. can be found Exercise 5D may now be attempted. The checkup exercise may now be attempted. Mathematics Support Materials: Mathematics 2 (Int 2) - Staff Notes 16 SIMULTANEOUS LINEAR EQUATIONS By the end of this set of exercises, you should be able to (a) Construct formulae to describe a linear relationship (b) Understand the significance of the point of intersection of two graphs (c) Solve simultaneous linear equations in two variables graphically (d) Solve simultaneous linear equations in two variables algebraically Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 17 SIMULTANEOUS LINEAL EQUATIONS A . Construction of Formula Exercise 1 1. A greengrocer sells Brussel Sprouts in 3 kilogram bags. The table compares the number of bags with the weight of sprouts sold. Number of Bags (N) Weight of sprouts (W) 1 3 2 6 3 4 5 6 9 12 15 18 Weight in kg (a) Copy and complete: Weight = ...... x No. Bags (b) Write a formula for the weight of sprouts. (c) Use your formula to find the weight of sprouts in 10 bags. (d) In your jotter, use your table to plot and join the points on a coordinate diagram like this :Ð (e) Extend your graph to show a straight line which passes through the origin. 21 18 15 12 9 6 3 0 1 2 3 4 5 6 Bags 2. A confectioner sells jelly eels in packs of ten. (a) Copy and complete the table: (b) (c) (d) (e) (f) Number of packs (P) 1 Number of eels (E) 10 2 3 4 5 6 Copy and complete :Ð Number of eels = ...... x No. packs Write a formula for calculating the number of eels. Use your formula to find the number of eels in 9 packs. Use your table to plot and join the points on a coordinate diagram. Extend your graph to show a straight line which passes through the origin. 3. The graph shows cooking times for roast beef. (a) Copy and complete the table: 1 Time (T) 20 2 3 4 5 6 (b) Write a formula for the time (T) taken to cook a roast if you know its weight (W). (c) Use your formula to find the time taken to cook a 10 pound roast . Time (min) Weight (W) 100 90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 weight (pound) Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 18 4. Mr. R. Highet called out Computer Fix to repair his computer. They have a Ôcall outÕ charge of £25 plus a charge of £8 per hour. No. Hours (h) 1 2 3 Charge £ (C) 33 41 49 4 5 (a) How much do Computer Fix charge for: (i) 4 hours? (ii) 5 hours? (b) Write a formula for the charge (C), given the number of hours worked (h). 5. To hire a cement mixer it costs a basic £8 plus £4 for each day you have the machine. (a) Copy and complete the table: No. Days (D) 1 Charge £ (C) 12 2 3 Cost (£Õs) (b) Write a formula for the charge (C) given the number of days (D) for which you have the machine. (c) In your jotter, use your table to plot and join the points on a coordinate diagram like this: (d) Extend your graph to cut the vertical (C) axis and give the coordinates of the point where the line cuts that axis. (e) Explain this point in relation to hiring a cement mixer. 4 5 24 20 16 12 8 4 0 1 2 3 4 5 6 Days 6. The graph shows defrosting times for a chicken. Weight (W pounds) 1 2 Time (T min) 15 3 4 5 6 time (mins) (a) Using the graph, copy and complete the table. 40 30 20 10 (b) Write a formula for the time (T) taken to cook a chicken if you know its weight (W). (c) Use your formula to find the time taken to cook a 10 pound chicken . 0 1 2 3 4 5 weight (pounds) Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 19 7. Fast Delivery charges £50, plus £5 per kilometre to deliver parcels. 100 90 80 70 60 50 40 30 20 10 Cost (£) (a) Write down a formula for the charge £C for a delivery of k kilometres. (b) Calculate the charge for a 10 kilometre trip. (c) Draw a graph of charges up to 10km, using these scales. COPY 0 2 4 6 8 10 8 10 kilometres 250 200 Wages (£) 8. Mrs. Divers sells cosmetics. She gets paid a basic £80 per week plus £10 each time she sells a product from the new Opius Perfume range. (a) Write down a formula for her wage £W for a week in which she sells P products. 150 COPY 100 (b) Work out her wage for a for a week in which she sells 20 products. 50 (c) Draw a graph of her wages for up to 20 products, using these scales. 0 2 4 6 No. of products 9. Mr. McGarrill, the school janitor, is ordering sweeping brushes at £10 each. If he pays quickly he finds that he can get a discount of £5 off his total bill. (a) Copy and complete the table: No. Brushes (B) 1 2 Cost £ (C) 5 15 25 (b) What is his bill for: (i) 4 brushes? 3 4 5 (ii) 5 brushes? (c) Write a formula for the cost (C) for a number of brushes (B). Cost (£) (d) In your jotter, use your table to plot and join the points on a coordinate diagram like this: 30 25 20 15 10 5 0 1 2 3 4 5 6 No. of brushes Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 20 10. A group of adults are having a night out at a tenÐpin bowling alley. The cost is normally £6 each, but a midweek special is giving £4 off the total bill. (a) Make up a table to show the total bill for 1, 2, 3, 4, 5, 6 bowlers. (b) Write a formula for the total bill (£T) for a number of bowlers (B). 30 Bill (£) (c) In your jotter, use your table to plot and join the points on a coordinate diagram like this: 20 10 0 1 2 3 4 5 6 Bowlers Revision:- Drawing Straight Lines Exercise 2 For each of the following equations of a straight line: ¥ choose three points on the line ¥ plot the points on squared paper, each one on a separate diagram ¥ draw a straight line through them. 1. y = x 5. y = 2x Ð 1 9. x + y = 6 2. y = 3x 6. y = 2 Ð x 10. x Ð y = Ð2 3. y = x + 1 7. y = 5 11. 2x + y = 0 4. y = 2x + 3 8. x = 3 12. y = Ðx + 1 B . Solving Simultaneous Linear Equations Graphically Exercise 3 By drawing the graphs represented by the following equations on squared paper, solve each pair of simultaneous equations. 1. x + y = 6 y=x 2. x+y=4 x + 2y = 6 3. xÐy=4 x Ð 2y = 6 4. x + y = 8 xÐy=2 5. x + 2y = 5 x Ð y = Ð1 6. y=x+2 y = Ðx Ð 4 7. x + 3y = 7 x Ð 3y = 1 8. y = 2x + 2 y = Ðx Ð 4 9. 2x Ð y = 3 y=5 11. 3x Ð 3y = Ð6 3x Ð 2y = 0 12. x + 3y = 8 2x Ð y = Ð5 10. 2x + y = 4 3x + 2y = 9 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 21 Exercise 4A 1. HenryÕs Rent a Car £20 per day GoudieÕs Car Hire £40 Deposit + £10 a day (a) Copy and complete the tables showing the charges for the two car hire companies. Number of days Cost (£) GoudieÕs 0 1 2 3 40 50 60 4 5 6 7 Number of days Cost (£) HenryÕs 0 1 2 3 0 20 40 4 5 6 7 (c) The two companies charge the same amount only once. For how many days is this? (d) Up to how many days is HenryÕs cheaper? Cost (£) (b) Draw the straight line graph for both car hire companies on the same coordinate diagram. 60 40 20 0 1 2 3 4 5 6 2. ÔHire a bike in Millport.Õ Mr. Dawes charges £1 deposit plus 50p per hour. Mr. Beckham charges No deposit, £1 per hour. Cost (£) (a) Make two tables to show the prices for up to 6 hours hire at DawesÕ and BeckhamÕs. (b) Draw the straight line graph for both bicycle hire companies on the same coordinate diagram. (c) For what number of hours hire is the cost the same at both shops? (d) If you wanted to hire a bike for 4 hours, which shop would you go to in order to save money? No. of days 6 5 4 3 2 1 0 1 2 3 4 5 6 3. RENT A COMPUTER are offering computers for £20 deposit, plus £5 per month. COMPU HIRE are offering similar computers for £10 per month, with no deposit. (a) Make two tables to show the prices for up to 5 months at each place. (b) Draw the straight line graph for both computer rental companies on the same coordinate diagram. (c) (i) For what number of months is the cost the same at both shops? (ii) What price is this? Cost (£) No. of hours 30 25 20 15 10 5 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 0 1 2 3 4 5 6 Months 22 (a) Make two tables to show the prices for up to a 10 mile journey at both firms. (b) Draw the straight line graph for both taxi companies on the same coordinate diagram. (c) For how many miles is the cost the same at both firms? (d) You are travelling only 2 or 3 miles Ð which taxi company would you phone to save money? Price (£Õs) 4. BLACK CAB TAXI COMPANY charge 50p per mile. RED TAXIS charge £2 for any journey up to 4 miles, then £1 per mile for each additional mile. 8 1 0 10 miles Exercise 4B 1. One adult and one child paid £8 to attend this football match. x+y=8 Two adults and one child paid £13. Third Lanark v Leith Athletic Adult Charge £x Child Charge £y (a) (b) (c) (d) 2x + y = 13 Draw the lines x + y = 8 and 2x + y = 13 on the same coordinate diagram using suitable points on each line. Write down the coordinates of the point of intersection. What is significant about this point in terms of prices to get into the match? What was the charge for 10 adults and 10 children at this match? 2. The professional at Worthwent Golf Club prices her goods as follows: Golf Balls £x Golf Gloves £y Arnold bought 2 golf balls and 1 golf glove for £8. Tiger bought 4 golf balls and 1 golf glove for £12. 2x + y = 8 4x + y = 12 (a) Draw the lines 2x + y = 8 and 4x + y = 12 on the same coordinate diagram using suitable points on each line. (b) Write down the coordinates of the point of intersection. (c) What was the cost of a golf ball? (d) What was the cost of a golf glove? (e) What does the professional charge for 3 golf balls and 3 golf gloves? 3. 2 jotters and 2 pencils cost 80p. 1 jotter and 3 pencils cost 60p. Let the cost of a jotter be x pence and the cost of a pencil be y pence. One equation from the data given is 2x + 2y = 80. (a) Write down the other equation in terms of x and y. (b) Draw the two straight lines which the equations represent on the same coordinate diagram using suitable points on each line. (c) Use your graph to find the cost of a jotter and the cost of a pencil. Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 23 4. 1 packet of Weedo and 1 packet of slug pellets costs £5. 1 packet of Weedo and 3 packets of slug pellets costs £9. Let the cost of a packet of Weedo be £x and the cost of a packet of slug pellets be £y. (a) Write down two equations in terms of x and y. (b) Draw the two straight lines which the equations represent on the same coordinate diagram using suitable points on each line. (c) Use your graph to find the cost of a packet of Weedo and the cost of a bottle of slug pellets. 5. Mary bought 3 TÐshirts and 2 bottles of colour dye for £12. Sally bought 2 of the TÐshirts and 5 bottles of colour dye for £30. Let the cost of a TÐshirt be £x and the cost of a bottle of colour dye be £y. (a) Write down two equations in terms of x and y. (b) Draw the two straight lines which the equations represent on the same coordinate diagram using suitable points on each line. (c) Use your graph to find the cost of a TÐshirt and the cost of a bottle of colour dye. 6. The total cost of two books is £10 and the difference in their cost is £2. Let the cost of a one book be £x and the cost of the other book be £y . (a) Write down two equations in terms of x and y. (b) Draw the two straight lines which the equations represent on the same coordinate diagram using suitable points on each line. (c) Use your graph to find the cost of each book. C . Solving Simultaneous Linear Equations Algebraically Exercise 5A Solve these simultaneous equations by eliminating x or y, etc. 1. x + y = 12 xÐy=8 2 x+y=6 xÐy=4 3. x + y = 10 xÐy=8 4. x + 2y = 6 x Ð 2y = 2 5. a + 4d = 9 a Ð 4d = 1 6. 3r + t = 10 3r Ð t = 2 7. 5p + q = 4 2p + q = 1 8. 6u + 6w = 6 4u + 6w = 6 9. 7x Ð 3y = 1 4x Ð 3y = Ð2 11. 5e Ð 2f = 8 Ðe + 2f = 0 12. Ð3x Ð 4y = 3 3x + y = 6 10. 4g Ð 5h = 13 3g Ð 5h = 11 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 24 Exercise 5B Solve these simultaneous equations by first multiplying both sides of the equations by suitable numbers. 1. x + 2y = 4 2x Ð y = 3 2. 3a + d = 9 a Ð 2d = 3 3. 4e Ð f = 11 e + 2f = 5 4. g + 2h = 7 2g Ð h = 9 5. m + 3n = 2 2m Ð n = 4 6. 5p + q = 3 p Ð 2q = 5 7. 3r + 2s = 1 r+s=0 8. 4t + 2u = 4 t+u=0 9. 3v Ð 4w = 13 v+w=2 10. x Ð y = 4 3x Ð 2y = 8 11. 5x Ð 2y = Ð1 x Ð 3y = 5 12. x Ð 3y = 1 2x Ð y = 7 Exercise 5C Solve these simultaneous equations by first multiplying both sides of the equations by suitable numbers. 1. 2p Ð 3q = 1 3p + 2q = 8 2. 2x + 4y = 14 7x + 3y = 27 3. 2v + 3w = 0 vÐw=5 4. 7a + 4d = 1 5a + 2d = Ð1 5. 2r Ð 3s = 12 3r Ð 2s = 13 6. 5x Ð 8y = 0 4x Ð 3y = Ð17 7. 3g + 2h Ð 6 = 0 gÐhÐ1=1 8. 3m + 5n Ð 23 = 0 5m + 2n Ð 13 = 0 9. 3f Ð 5g Ð 11 = 2 2f + 4g Ð 9 = 7 Exercise 5D Write down a pair of simultaneous equations for each picture, then solve them to answer the question. (Use £x and £y to represent the cost of one of each item each time). 1. Total cost £9 Find the cost of: Total cost £5 (a) one ice cream sundae. (b) one mug of cocoa. 2. Total cost £24 Find the cost of: (a) one hammer. Total cost £21 (b) one spanner. Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 25 3. Total cost 55p Find the cost of: (a) one apple. Total cost 75p (b) one pear. 4. Total cost £3á50 Find the cost of: (a) one frothy drink. Total cost £2á50 (b) one slice of cake. 5. Total cost £90 Find the cost of: (a) one football. Total cost £110 (b) one rugby ball. 6. Total cost £7 Find the cost of: (a) one disk. Total cost £6á50 (b) one calculator. 7. Total cost £6á50 Find the cost of: (a) one hot dog. Total cost £7 (b) one hamburger. 8. At a supermarket, a lady paid £2á70 for 6 red peppers and 5 corn on the cobs. At the same supermarket, a man paid £1á20 for 3 red peppers and 2 corn on the cobs. Find the cost of: (a) one pepper. (b) one corn stick. Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 26 9. At a newsagent, a boy paid £1á10 for 2 memo pads and 7 pencils. At the same shop, a girl paid £1á60 for 7 memo pads and 2 pencils. Find the cost of: (a) one memo pad. (b) one pencil. 10. An adultÕs ticket for the cinema is £3 more than a childÕs. The adultÕs ticket is also twice that of the childÕs. Let the price of an adultÕs ticket be £x and the price of a childÕs ticket be £y. Form a pair of simultaneous equations and solve them to find the price of each ticket. Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 27 CHECKUP FOR SIMULTANEOUS LINEAR EQUATIONS (a) Copy and complete the table: Weight (W) Time (T) 0 2 4 6 8 10 12 14 16 18 20 0 4 (b) Write a formula for the time (T) taken to defrost a turkey if you know its weight (W). (c) Use your formula to find the time taken to defrost a 15 pound turkey. Time (hrs) 1. The graph shows defrosting times at room temperature for Christmas turkey. 40 36 32 28 24 20 16 12 8 4 0 2 4 6 8 10 12 14 16 18 20 weight (pound) Cost (pence) 2. Pizza Point will deliver pizzas to your door. The charge is 50p, plus 10p per mile. (a) Write down a formula for the charge C pence for a delivery of M miles. 100 (b) Work out the charge for a 5 mile delivery. (c) Draw a graph of charges up to 5 miles, using the scales shown. (d) What would be the charge for a 10 mile delivery ? 80 60 40 20 0 1 2 3 4 5 Miles 3. By drawing graphs of these equations on squared paper, solve each pair of simultaneous equations. (b) x + 2y = 7 4x Ð y = 10 4. HIGH FLY offer balloon trips at £10 basic, plus £2 per kilometre travelled. FLIGHT BALLOONS offer the same trips at £4 per kilometre, with no other charges. (a) Make two tables to show the prices for up to a trip of 6 km with both companies. (b) Draw the straight line graph for both companies on the same coordinate diagram. (c) (i) How many kilometres can you travel for the same price at both businesses? (ii) What price is this? (c) Cost (£Õs) (a) x + y = 8 y=x x + 3y = 0 x Ð 2y = 5 24 20 16 12 8 4 0 1 2 3 4 5 6 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials kilometres 28 5. Terry bought a bottle of shampoo and a bottle of conditioner for £6. Lesley bought 4 bottles of shampoo and a bottle of conditioner for £12. Let the cost of a bottle of shampoo be £x and the cost of a bottle of conditioner be £y. (a) Write down two equations in terms of x and y. (b) Draw the two straight lines which the equations represent on the same coordinate diagram using suitable points on each line. (c) Use your graph to find the cost of a bottle of shampoo and the cost of a bottle of conditioner. 6. Solve these simultaneous equations algebraically: (a) x + y = 20 (b) x Ð 3y = Ð1 xÐy=4 x + 3y = 11 (d) v + 3w = 7 2v Ð w = 0 (e) 2p + 3q = 19 4p Ð 7q = Ð27 (g) 5s + 3t = 19 7s Ð 2t = 8 (h) 4x Ð 3y Ð 1 = 4 3x + 4y Ð 10 = 0 (c) 2x + y = 10 Ð2x + y = Ð10 (f) 2x Ð 3y = 1 3x + 2y = Ð5 7 . Write down a pair of simultaneous equations for each picture, then solve them to answer the question. (Use £x and £y to represent the cost of one of each item). (a) Total cost £36 Find the cost of: (i) one spider. Total cost £28 (ii) one turtle. (b) 5 pairs of compasses and 2 pairs of scissors together cost £2á30. 3 pairs of compasses along with 3 pairs of scissors cost £2á10. Find the cost of: (i) one pair of compasses. (ii) one pair of scissors . 8. The sum of two whole numbers is 112, and their difference is 36. Form a pair of simultaneous equations and solve them to find the two numbers. Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 29 Exercise 4B 1. 2. 3. 4. 5. 87á3° (a) 40á8° (b) 83á9° (c) 47° (d) 61á7° 100á3° (a) 109á5° (b) 111á8° (c) 113á3° x = 40á1°, y = 57á4° (e) 54á0° (f) 26á7° (b) Area = 214á8 cm2 6. (a) x = 56á9° Checkup for Trigonometry 1. (a) Ð0á342 (b) Ð0á839 (c) Ð0á087 2. 91.6 cm2 3. 50á7 cm2 4. (a) 11á4 cm (f) 22á2 cm 5. 118á7° 6. 94á5 km 7. 199 m (b) 11á1 cm (c) 12á5 cm (d) 9á3 cm (g) 80á4° (h) 72á3° (i) 131á6° (e) 8á9 cm 8. 117á9 cm2 Simultaneous Linear Equations Exercise 1 1. (a) 3 (b) W = 3N (c) 30kg (d)(e) 2. (a) 1/10 2/20 3/30 4/40 5/50 6/60 in table (e) (f) (b) 10 (c) E = 10P 3. (a) 1/20 2/40 3/60 4/80 5/100 6/120 in table (b) T = 20W 4. (a) £57 £65 (b) C = 8h + 25 5. (a) 1/12 2/16 3/20 4/24 5/28 in table (b) C = 4D + 8 (c) (d) (0,8) (e) Costs £8 before even paying for any days !! 6. (a) 1/15 2/25 3/35 4/45 5/55 6/65 in table (b) T = 10W + 5 (c) 105 mins 7. (a) C = 5k + 50 (b) £100 (c) 50 8. (a) W = 10P + 80 (b) £280 (d) 90 (c) 200 mins 8 (c) 80 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 33 9. (a) 1/5 2/15 3/25 4/35 5/45 in table (b) £35; £45 (c) C = 10B Ð 5 (d) Ð5 10. (a) 1/2 2/8 3/14 4/20 5/26 6/32 in table (b) T = 6B Ð 4 (c) Ð4 Exercise 2 1. Graph of a straight line through (0,0), (1,1) (2,2) etc. 2. Graph of a straight line through (0,0), (1,3) (2,6) etc. 3. Graph of a straight line through (0,1), (1,2) (2,3) etc. 4. Graph of a straight line through (0,3), (1,5) (2,7) etc. 5. Graph of a straight line through (0,Ð1), (1,1) (2,3) etc. 6. Graph of a straight line through (0,2), (1,1) (2,0) etc. 7. Graph of a straight line through (0,5), (1,5) (2,5) etc. 8. Graph of a straight line through (3,0), (3,1) (3,2) etc. 9. Graph of a straight line through (0,6), (1,5) (2,4) etc. 10. Graph of a straight line through (0,2), (1,3) (2,4) etc. 11. Graph of a straight line through (0,0), (1,Ð2) (2,Ð4) etc. 12. Graph of a straight line through (0,1), (1,0) (2,Ð1) etc. Exercise 3 1. (3,3) 2. (2,2) 8. (Ð2,Ð2) 9. (4,5) 3. (2,Ð2) 10. (Ð1,6) 4. (5,3) 11. (4,6) 5. (1,2) 12. (Ð1,3) 6. (Ð3,Ð1) 7. (4,1) Exercise 4A 1. (a) Goudies 0/40 1/50 2/60 3/70 4/80 5/90 6/100 7/110 HenryÕs 0/0 1/20 2/40 3/60 4/80 5/100 6/120 7/140 (b) Straight lines crossing at (4,80) (c) 4 days (d) 3 days 2. (a) Dawes 0/1 1/1á50 2/2 3/2á50 4/3 5/3á50 6/4 Beckams 0/0 1/1 2/2 3/3 4/4 5/5 6/6 (b) Straight lines crossing at (2,2) (c) 2 hours (d) Dawes 3. (a) Rent a Computer 0/20 1/25 2/30 3/35 4/40 5/45 Compu Hire 0/0 1/10 2/20 3/30 4/40 5/50 (b) Straight lines crossing at (4,40) (c) 4 £40 4. (a) Black 0/0 1/0á5 2/1 3/1á5 4/2 5/2á5 6/3 7/3á5 8/4 9/4á5 10/5 Red 0/2 1/2 2/2 3/2 4/2 5/3 6/4 7/5 8/6 9/7 10/8 (b) Lines crossing at (4,2) Red (c) 4 miles (d) Black Cab Black Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 34 Exercise 4B 1. 2. 3. 4. 5. 6. (a)(b) Straight lines crossing at (5,3) (c) £5 adult £3 child (d) £80 (a)(b) Straight lines crossing at (2,4) (c) £2 (d) £4 (e) £18 (a) x + 3y = 60 (b) Straight lines crossing at (30,10) (c) jotter 30p pencil 10p (a) x + y = 5 x + 3y = 9 (b) Straight lines crossing at (3,2) (c) Weedo £3 slug £2 (a) 3x + 2y = 12 2x + 5y = 30 (b) Straight lines crossing at (0,6) (c) shirt free dye £6 (a) x + y = 10 x Ð y = 2 (b) Straight lines crossing at (6,4) (c) £6 and £4 Exercise 5A 1. (10,2) 6. (2,4) 11. (2,1) 2. (5,1) 7. (1,Ð1) 12. (3,Ð3) 3. (9,1) 8. (0,1) 4. (4,1) 9. (1,2) 5. (5,1) 10. (2,Ð1) 2. (3,0) 7. (1,Ð1) 12. (4,1) 3. (3,1) 8. (2,Ð2) 4. (5,1) 9. (3,Ð1) 5. (2,0) 10. (0,Ð4) 2. (3,2) 7. (2,0) 3. (3,Ð2) 8. (1,4) 4. (Ð1,2) 9. (6,1) 5. (3,Ð2) ice cream £2 hammer £6 apple 20p drink £1á50 football £10 disk 50p hot dog £1 pepper 20p pad 20p adult £6 cocoa £1 spanner £3 pear 15p cake 50p rugby ball £20 calculator £2 hamburger £1á50 corn 30p pencil 10p child £3 Exercise 5B 1. (2,1) 6. (1,Ð2) 11. (Ð1,Ð2) Exercise 5C 1. (2,1) 6. (Ð8,Ð5) Exercise 5D 1. 4x + y = 9 2x + y = 5 2. 3x + 2y = 24 2x + 3y = 21 3. 2x + y = 55 3x + y = 75 4. 2x + y = 3á50 x + 2y = 2á50 5. 5x + 2y = 90 5x + 3y = 110 6. 2x + 3y = 7 5x + 2y = 6á50 7. 3x + 2y = 6á50 2x + 4y = 7 8. 6x + 5y = 2á70 3x + 2y = 1á20 9. 2x + 7y = 1á10 7x + 2y = 1á60 10. x Ð y = 3 x = 2y or equivalent Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 35 Checkup for Simultaneous Linear Equations 1. (a) 0/0 2/4 4/8 6/12 8/16 10/20 12/24 14/28 16/32 18/36 20/40 in table (b) T = 2W (c) 30 hours 2. (a) C = 10M + 50 (b) 100p (c) (d) 150p 50 3. (a) (4,4) (b) (3,2) (c) (3,Ð1) 4. (a) High Fly 0/10 1/12 2/14 3/16 4/18 5/20 6/22 in table Flight Balloons 0/0 1/4 2/8 3/12 4/16 5/20 6/24 in table (b) Straight lines crossing at (5,20) (c) 5km £20 5. (a) x + y = 6 4x + y = 12 (b) Straight lines crossing at (2,4) (c) Sham £2 Cond £4 6. (a) (12,8) (b) (5,2) (c) (5,0) (d) (1,2) (e) (2,5) (f) (Ð1,Ð1) (g) (2,3) (h) (2,1) 7. (a) 3x + y = 36 2x + y = 28 spider £8 turtle £12 (b) 5x + 2y = 2á30 3x + 3y = 2á10 compasses 30p scissors 40p 8. 74 & 38 Specimen Assessment Questions 1. 25á4 cm2 2. 3. 4. 5. (a) 16á1 cm (b) 26á2° (c) 7á1 cm 124 m 58á0° (a) £50 £70 £90 £110 £130 (b) C = 20h + 30 6. 7. 8. 9. (a) (a) (a) (a) (c) 30 (6,4) (b) (20,30) x + y = 4 2x + 4y = 10 (b) Straight lines crossing at (3,1) (c) Adult £3 Child £1 (1,Ð1) (b) (5,2) (c) (4,2) 3x + y = 2á60 x + 2y = 2á20 coke 60p chips 80p Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 36

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