Simultaneous linear equations

Transcription

1 8 Stretch lesson: Simultaneous linear equations Stretch objectives Before ou start this chapter, mark how confident ou feel about each of the statements below: I can write simultaneous equations to represent a situation. I can solve two linear simultaneous equations algebraicall. I can solve two linear simultaneous equations graphicall. I can solve simultaneous linear equations representing a real-life situation and interpret the solution. Check-in questions Complete these questions to assess how much ou remember about each topic. Then mark our work using the answers at the end of the lesson. If ou score well on all sections, ou can go straight to the Revision Checklist and Eam-stle Questions at the end of the lesson. If ou don t score well, go to the lesson section indicated and work through the eamples and practice questions there. Solve these simultaneous equations. Go to 8. a b + 7a = b a - b = 9 b + a = a + b = The graphs of + = and = + are plotted on the same aes. Use the graphs to solve this pair of simultaneous equations. Go to 8. = + + = 7 8 Sahid has lots of sports equipment in different bags. One bag contains two footballs and three cricket balls, which have a total mass of g. A second bag contains three footballs and two cricket balls, which have a total mass of g. Work out the mass of one cricket ball. Go to 8. Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7

2 8. Solving simultaneous equations algebraicall Simultaneous equations in two variables are equations that are both true for the same pair of variables. You can solve simultanous equations using algebraic methods or b using a graph. In straightforward eamples, the coefficients of one of the variables will be the same in both equations so ou can eliminate one of the variables b adding or subtracting the equations. Otherwise ou will need to multipl one or both of the original equations b a constant before ou can use the elimination method. The eample shows how to use the elimination method when there are no matching coefficients. Eample Q A Solve this pair of simultaneous equations. + = 8 - = + = 8 () - = () + = () - 9 = () First, label the equations. Since no coefficients match, multipl equation () b and equation () b and rename them () and (). The coefficient of in equations () and () is the same, so ou can now subtract them and solve to find the value of. + = - - (-9) = + 9 = - = - Substitute the value of (= -) into equation () to find the value of. + (-) = 8 You could also substitute into equation (). + (-) = 8 = 8 + = = Check in equation (): ( ) - ( -) = So, the solution is =, = -. Eam tips Alwas check that our solutions are correct b substituting them into the other equation. Practice questions Solve these simultaneous equations using algebra. Remember to check our answers. a + = b + = c + = 8 - = + = - = - Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7

3 Solve the simultaneous equations algebraicall. a + = b + = c + = + = + = = Solve the pairs of simultaneous equations using the elimination method. a + = b + = c - = + = + = 9 + = Solve these simultaneous equations. a - = b - = c - = + = - + = - + = Solve each pair of simultaneous equations. a p + q = b + =. c - =. p - q = 7 - = =. 8. Solving simultaneous equations graphicall When ou plot two graphs on the same pair of aes, the point of intersection (where the cross) represents the solution of the simultaneous equations. Eample Q Solve these simultaneous equations graphicall. + = + = A Draw the graph of + =. When =, = so = : (, ) When =, = so = : (, ) Draw the graph of + =. When =, = : (, ) When =, = : (, ) The lines intersect at (-, ). The simultaneous solution of the equations is = -, =. Point of intersection at (, ) = + = Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7

4 Practice questions The graph of = + is shown on the aes. Set up a table of values for the equation = -, then draw the graph on a cop of the aes. Using our graph, find the solution of the simultaneous equations. 7 = The graph of = - is shown on the aes. Draw the graph of = + on a cop of the grid. Using our graph, find the simultaneous solution of the equations. 8 = 8 8 Draw the graphs of - = and - = on the same aes. Solve the equations simultaneousl b reading the coordinates of the point of intersection from our graph. Hint Rearrange each equation before ou set up a table of values. Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7

5 Solve each pair of simultaneous equations graphicall. a = + b = = = + Solve these simultaneous equations graphicall. a + = b + = c - = - = - = + = 7 8. Solving problems using simultaneous equations Simple real-life problems can be solved b forming and solving a pair of simultaneous equations. Eample Q A Trac and Ian are organising a part. Trac paid for five hats and four balloons from the to shop. She found that she didn t have enough, so Ian went back and bought three of the same hats and five of the same balloons. He paid. Work out the cost of one hat and the cost of one balloon. Use h to represent the cost of a hat and b to represent the cost of a balloon. First set up two equations and label them. h + b = () h + b = () Multipl, since none of the coefficients are the same. h + b = () () h + b = () () Subtract to eliminate the h. b = 9 () - () b = Substitute b = into (). h + ( ) = h = h = Check in (). + = So one hat costs and one balloon costs. Eam tips When setting up our own simultaneous equations, use appropriate letters for each of the items, for eample, c for cakes and m for muffins. Remember to interpret our results at the end of the question. Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7

6 Practice questions A café sells a total of cakes and muffins for 87. The cost of a cake is. and the cost of a muffin is.. How man cakes did the sell? Mand is selling hot dogs and burgers at a village fair. Tom pas 8. for three hot dogs and two burgers. Mick pas 8 for four hot dogs and one burger. a Find the cost of one hot dog. b Find the cost of one burger. Anne sells homemade shawls and scarves. She sells one shawl and two scarves for 7. She sells two shawls and si scarves for. How much does one scarf cost? Elliot does MOTs and fits brake pads. He is writing his schedule for the week. a How much time does Elliot allow to fit one set of brake pads? b How long will Mrs Thomson have to wait for Elliot to fit one set of brake pads and MOT her car? Liz runs a dog walking service. She offers individual short walks or long walks. Two short walks and three long walks take her hours minutes. Three short walks and five long walks take her hours and minutes. How long will she need for one short walk and four long walks? REVISION CHECKLIST When ou solve a pair of linear simultaneous equations, ou need to find the values of both unknown variables. You can use the elimination method to eliminate one of the variables in a pair of simultaneous equations. The coordinates of the point of intersection of two graphs give the solution of the simultaneous equations. Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7

7 Eam-stle questions Draw graphs to solve the simultaneous equations + = 7 and - =. The graph of = - is plotted on the aes below. = Use the graph to help ou solve the simultaneous equations - = and + =. The graph of = + is shown on the aes = Use the graph to help ou solve the simultaneous equations - = - and + = 9. Solve the simultaneous equations = + and + = 7 graphicall. Solve the simultaneous equations + = 8 and + = graphicall. Solve these simultaneous equations. + = + = 7 The sum of two numbers is. The difference between the same two numbers is. What are the two numbers? 8 Solve these simultaneous equations. + = 9 + = Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7

8 9 The sum of two numbers is. The difference between the same two numbers is 7. What are the two numbers? Georgia bus two cans of cola and a chocolate bar for.. Marcus bus three cans of cola and two chocolate bars for.. How much will Jen pa for one can of cola and four chocolate bars? Solve these simultaneous equations. + = - = Sami bus DVDs and CD for.8. Dougie bus DVDs and CDs for.. How much will Angie pa for DVDs and CDs? Solve these simultaneous equations. + = - = The mean of two numbers is 8. The difference between the same two numbers is. Find the two numbers. Harriet is selling second-hand books. She sells four paperback books and three hardback books for.. She sells seven paperback books and eight hardback books for.. How much will she charge for five paperback books and four hardback books? Iannis teaches music. On Wednesda he teaches five guitar lessons and two theor lessons; this takes hour minutes. On Thursda he teaches three guitar lessons and ten theor lessons; this takes hours minutes. On Frida he epects to teach four guitar lessons and si theor lessons. How much time will this take? 7 The mean of two numbers is 9. The difference between the same two numbers is. Find the two numbers. 8 Bernice paid. for three large candles and eight small candles. Pauline paid. for two large candles and twelve small candles. How much will Claire pa for five large candles and two small candles? 9 Gunna works in a nursing home. She stles hair and paints nails for the female residents. On Monda, she stles three lots of hair and paints five sets of nails. This takes her hours minutes. On Tuesda, she stles five lots of hair and paints three sets of nails. This takes her hours. She has been asked to stle two lots of hair and paint four sets of nails on Thursda. If she arrives at the nursing home at 9 a.m., at what time can she epect to leave? Alan works in a print shop. It takes minutes seconds to print flers and leaflets. It takes 9 minutes seconds to print flers and leaflets. How much time will it take to print flers and 7 leaflets? Now go back to the list of objectives at the start of this chapter. How confident do ou now feel about each of them? Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7

9 Chapter 8 Stretch lesson: Answers Check-in questions a a =, b =. b a =, b = =, = g 8. Solving simultaneous equations algebraicall a =, = b =, = c =, = a =., = b =, =. c =, = a =, = b =, = c =, = a =, = b =, = c =, = a p =, q = b =., =. c =.7, =. 8. Solving simultaneous equations graphicall =, = =, = 7 =, = Eam-stle questions =, = =, = 9 =, = =, = =, = =, = 7 and 8 8 =., =. 9 8 and.8 =, =. =., = and. hours 7 8 and : p.m. 7 minutes seconds a =, = b =, = a =, = b =, = c = 7, = 7 8. Solving problems using simultaneous equations cakes Cost of one hot dog., cost of one burger. a minutes b hour minutes hours minutes Edecel GCSE (9-) Maths for Post- HarperCollinsPublishers 7