* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Simultaneous Equations - Pearson Schools and FE Colleges
Quartic function wikipedia , lookup
Cubic function wikipedia , lookup
Quadratic equation wikipedia , lookup
Signal-flow graph wikipedia , lookup
Elementary algebra wikipedia , lookup
System of polynomial equations wikipedia , lookup
System of linear equations wikipedia , lookup
Get started 5 Simultaneous equations This unit will help you to set up and solve different types of simultaneous equations. AO1 Fluency check 1 Simplify b4y − y a −2x + 3x c 3x − x d−2y + (−2y) 2 Solve a 3x = 9 x= c x + 3 = 10 b2y = −8 y= x = d3 + 2y = −4 y = 3 Substitute y = 2 into 3x + 2y = 16 and solve to find the value of x. x= Number sense 4 Use the operations to complete each equation. x + y = 10 ×2 ×3 ×5 2x + 2y = x+ y = 50 3x + 3y = Key points Simultaneous equations are two (or more) equations that involve two (or more) letters. These 1 Solving simultaneous equations means finding the value of all the letters. skills boosts will help you to write and solve simultaneous equations. Subtracting to eliminate a variable 2 Adding to eliminate a variable 3 Multiplying an equation first 4 Setting up simultaneous equations You might have already done some work on simultaneous equations. Before starting the first skills boost, rate your confidence with these questions. 1 x + 2y = 7 x+y=4 2 3x + 5y = 32 x − 5y = −16 3 4 5x + 2y = 12 4x + y = 9 3 shirts and 2 hats cost £61. 2 shirts and 2 hats cost £46. How much is a shirt? How much is a hat? How confident are you? Unit 5 Simultaneous equations 31 Skills boost 1 Subtracting to eliminate a variable To solve a pair of simultaneous equations, add or subtract them to eliminate one of the variables. When there are two identical terms with the same sign, subtract one equation from the other. Guided practice Worked exam question Solve 2x + 3y = 9 2x + y = 7 Label one equation A and one equation B. 2x + 3y = 9 A 2x + y = 7 B 2y = 2 so y = 1 A Subtract B from A. 3y − y = B Solve the equation to find the value of y. y= ,y= x= 2 a Solve 4x – 6y = – 18 4x + 2y = 22 ,y= x= y y y x x y 7 6 x=3 Substitute y = into A. =9 2x + 2x = x=3 Solution x = 3, y = 1 2x + 4y = 14 2x + y = 5 x 0 A − B 1 a Solve x 9 1 Check your solutions work for equation B. 2 × 3 + 1 = 7 ✓ bSolve x + 3y = 10 x − y = 2 x = ,y= c Solve x + 5y = 7 3x + 5y = 11 x = bSolve 5x + 2y = 17 6x + 2y = 20 x = ,y= ,y= c Solve 2x + 3y = 5 2x + 4y = 8 x = ,y= Hint Simultaneous equations can have negative solutions. Exam-style question 3 Solve x + 3y = 11 4 a 3x – 2y = 10 x – 2y = 2 x= Reflect ,y= ,y= x= 2x + 3y = 16 b2x – y = 8 5x – y = 23 x = ,y= c 2x + y = 13 5x + y = 4 x = How do you decide which equation to subtract from the other? 32 Unit 5 Simultaneous equations (3 marks) ,y= Skills boost 2 Adding to eliminate a variable When two terms have the same coefficient and variable but different signs, add the equations to eliminate the variable. Guided practice Solve 2x – y = 11 x + y = 10 Subtracting does not eliminate a variable: 2x − y = 11 A x + y = 10 B Label one equation A and one equation B. 2x – y = 11 A x + y = 10 B A − B : x − 2y = 1 You still have x and y. The coefficients of y are the same and the signs are different, you can add the equations to eliminate the y variable. Add A and B and solve to find x. A + B 3x + 0 = 3x = x= Substitute x = 7 into B. + y = 10 y= Solution x = 7, y = 3 Check your solution works for equation A. 2 × 7 − 3 = 11 ✓ 1 a Solve bSolve x + 2y = 7 3x − 2y = 13 3x − y = 5 x + y = 7 ,y= x= x = ,y= c Solve 2x + 3y = 26 x – 3y = 4 x = ,y= Exam-style questions 2 Solve –x + 3y = 1 x – y = 3 x= ,y= (3 marks) x= ,y= (3 marks) 3 Solve x + 2y = 12 3x – 2y = 12 4 Solve a 2x – 3y = −15 b–4x + y = −9 c –x + 3y = 1 4x + 3y = 33 4x – 3y = −5 x = 3 + y x= ,y= x= ,y= x = ,y= Hint Rearrange the second equation so that x and y are on the same side of the equals sign. Reflect How do you decide whether to add or subtract the equations? Unit 5 Simultaneous equations 33 Skills boost 3 Multiplying an equation first If the x or y coefficients are not equal, multiply one or both of the equations so either the two x terms or the two y terms have the same coefficient. Guided practice Worked exam question Solve 2x + y = −1 x + 4y = 10 Label one equation A and one equation B. 2x + y = –1 A x + 4y = 10 B Multiply A by 4. x+ y = –4 Subtract B from 8x + 4y x + 4y 7x + 0 x The coefficient of y in equation A is 1. 1 × 4 = 4, so multiplying A by 4 makes the y coefficients in A and B equal. A ×4 4 × A and solve to find x. = –4 4 × A − = 10 B = –14 = –2 You could multiply B by 2 instead so that both equations have the same x coefficient. Substitute x = –2 into A. (2 × –2) + y = –1 + y = –1 y= Solution x = –2, y = 3 1 Solve 3x + 2y = 17 x + y = 7 x= Check your solutions work for equation B. −2 + (4 × 3) = 10 ✓ Hint Either multiply B by 3 (to make the x coefficients equal) or by 2 (to make the y coefficients equal). A B ,y= 2 Solve a 3x − y = 10 bx + 2y = −1 x + 2y = 8 −3x + y = 10 x= ,y= x = ,y= c 2x − 3y = −4 x + y = −7 x = ,y= 3 Solve a x + 2y = 0 b5x − y = 37 c 2x + −4y = 0 3x − y = −14 x − 3y = 13 x + 2y = 12 x= ,y= x = Hint Is it easier to multiply the first equation by 3 or the second equation by 2? 34 Unit 5 Simultaneous equations ,y= x = ,y= Skills boost 4 Solve a 10x − 5y = –55 b2x + 5y = 1 c 3x + 2y = 36 x − 2y = –19 –x – y = –5 2x – y = 17 x= ,y= ,y= x = ,y= Hint Multiply both equations so two terms have the 5 Solve 2x + 3y = 7 5x + 2y = 1 x= x = same coefficient. You could multiply A by 5 and B by 2. A B ,y= 6 Solve a 4x + 3y = –5 b3x + 7y = –4 c 6x + 2y = 10 3x + 5y = –1 2x + 5y = –2 5x + 10y = 100 x= ,y= x = ,y= x = ,y= 7 Solve a x + 2y = 3 b5x − y = 5 c x + 5y = −1 4x + 2y = 9 10x + 2y = −2 8x − 20y = 28 x= ,y= x = ,y= x = ,y= Hint Solutions to simultaneous equations can be fractions. 8 Solve a 8x + 2y = 2 b2x − 5y = −14 c −4x + 3y = 13 −4x − 3y = 3 −3x + 2y = −1 6x − 7y = −32 x= ,y= x = ,y= x = ,y= Exam-style question 9 Solve 4x + 2y = 5 8x − 3y = 24 x= Reflect ,y= (3 marks) How did you decide which equation to multiply? Unit 5 Simultaneous equations 35 Skills boost 4 Setting up simultaneous equations Guided practice 2 cookies and 3 sandwiches cost £8. 5 cookies and 1 sandwich cost £7. Find the cost of a a cookie ba sandwich. Write the equations using x and y. £8 Let x = cost of a cookie and y = cost of a sandwich. A x y y y 2 3 cookies sandwiches Label the equations A and B. A 2x + 3y = 8 5x + y = 7 B Multiply B by 3. y= 15x + x £7 B x x x x 5 cookies 3× B Subtract A from 3 × B and solve to find x. 15x + 3y = 21 3× B A − 2x + 3y = 8 13x + 0 = x= x y 1 sandwich 1 cookie costs £x 2 cookies cost £2x Substitute x = 1 into equation A and solve to find y. a A cookie costs £ b A sandwich costs £ 1 At a café 3 lemonades and 1 cola cost £5. 2 lemonades and 4 colas cost £10. Work out the cost of a a lemonade ba cola. 2 4 juices and 1 cake cost £3. 2 juices and 3 cakes cost £4. Work out the cost of a a juice ba cake. 3 Tickets for 2 adults and 3 students cost £38. The cost for 5 adults and 2 students is £62. Work out the price of a an adult ticket ba student ticket. 4 Entry for 2 adults and 5 children costs £41. Entry for 3 adults and 2 children costs £34. Work out the price of a adult entry bchild entry. Exam-style question 5 4 pens and 3 notebooks cost £17. 6 pens and 2 notebooks cost £15.50. Find the cost of a a pen Reflect ba notebook. Do you always need to use x and y when setting up equations? 36 Unit 5 Simultaneous equations (5 marks) Get back on track Practise the methods Answer this question to check where to start. Check up What is the simplest way to start solving these equations? a x + 5y = 11 b 5x + y = 7 A B Multiply a by 7 C Multiply b by 5 D Multiply a by 5 If you ticked B or C finish solving the equations. Then go to Q2. Subtract b from a If you ticked A or D go to Q1 for more practice. 1 Solve a 2x + y = 32 3x + y = 42 x= ,y= bx + 2y = 9 3x − 2y = 11 x = ,y= c 5x + 2y = 21 −5x + y = 3 x = ,y= 2 Solve a 2x + y = 10 x − 2y = −5 x= ,y= b3x − 2y = 18 x + 4y = 13 x = ,y= c 24x + 3y = 0 2x + y = −2 x = 3 Solve a 2x + 4y = 0 3x + 2y = 12 x= ,y= bx + 2y = 0 3x − y = −14 x = ,y= ,y= c 8x − y = 7 2x − 3y = 10 x = ,y= 4 Solve a 3x + 7y = 15 5x + 2y = −4 x= ,y= c 3x + 5y = 3 b 3x + 6y = 3 4x − 4y = −512x + 10y = −2 x = ,y= x = ,y= 5 At a fair the cost of a ride for 1 adult and 3 children is £11. The cost for 2 adults and 2 children is also £11. Work out the cost of a an adult ticket ba child ticket. Exam-style question 6 Solve 2x + 5y = −19 3x − 4y = 29 x= ,y= (3 marks) Unit 5 Simultaneous equations 37 Get back on track Problem-solve! 1 At the cinema tickets for 2 adults and 3 children cost £33. Tickets for 1 adult and 1 child cost £14. Write and solve a pair of simultaneous equations to find the cost of a an adult ticket ba child ticket. 2 x and y are two numbers. Their sum x + y equals 16. Their difference x − y equals 6. What are the values of x and y? x = ,y= 3 Find the two numbers whose difference is 15 and whose sum is 35. , Exam-style questions 4 Solve these simultaneous questions 4x = y + 7 10x – 2y = 15 x= ,y= (4 marks) h= ,b= (3 marks) 5 Jane’s wages are calculated using the formula Wages = number of hours, n × hourly rate, h + bonus, b W = nh + b When she works 30 hours her wage is £330. When she works 34 hours her wage is £370. What is Jane’s hourly rate and what is her bonus? 6 Write equations and solve to find the cost of these drinks and snacks. a 3 teas and 1 coffee cost £7. 2 teas and 4 coffees cost £13. b4 milkshakes and 1 cookie cost £8.20. 2 milkshakes and 3 cookies cost £7.10. 7 Three first class stamps and one second class stamp cost £2.47. One first class stamp and two second class stamps cost £1.74. What is the price difference between first and second class stamps? 8 Why is it not possible to solve these simultaneous equations? A 3x – 8y = 10 B6x – 16y = 20 Now that you have completed this unit, how confident do you feel? 1 Subtracting to eliminate a variable 2 Adding to eliminate a variable 38 Unit 5 Simultaneous equations 3 Multiplying an equation first 4 Setting up simultaneous equations