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Transcript
Key Stage 3 Number and Algebra Dimension Learning Unit: Linear Equations in Two Unknowns Learning Objectives: • plot and explore the graphs of linear equations in 2 unknowns • formulate and solve simultaneous equations by algebraic and graphical methods • be aware of the approximate nature of the graphical method Programme Title: Simultaneous Equations Programme Objectives 1. 2. 3. 4. 5. Recognize the graphs and solutions of linear equations in 2 unknowns. Understand the graphical solutions of simultaneous equations. Solve simultaneous equations by the method of substitution. Solve simultaneous equations by the method of elimination. Solve practical problems of simultaneous equations. Programme Content In considering the number of different gift boxes of tea, the idea of the solutions of linear equations in two unknowns is introduced. It points out that there is not a unique solution for a linear equation in two unknowns as the possible solutions are the dots on a straight line in the coordinate plane. The programme uses the inter-crossing station of the KCR and MTR route to bring forth the idea of the graphical solution of simultaneous equations. The unique solution is given by the point of intersection of the two straight line graphs. -1- In the discussion of solving a problem of mixing two kinds of tea, the procedures of solving problems of simultaneous equations are introduced: Step (1): Use letters to represent unknowns. Step (2): Use given conditions to form equations. Step (3): Solve equations (graphical, substitution or elimination). Step (4): Check solution. It also points out the approximate nature of the graphical solutions and thus introduces the algebraic methods, substitution and elimination, for solving simultaneous equations to obtain accurate solutions. Worksheet Answers 1. 6; solution is: y 3.75 and x 1.25; the solution is the same as that of the programme. 2. It is more convenient to substitute y from equation (2) into equation (1) so as to avoid fractional expressions, or to eliminate y to form a linear equation in x so as to avoid multiplying both equations by numbers. (some of the possible answers). Solution is: x 5 and y 2. 3. Let Kitty had x correct answers and y wrong answers. therefore, xy 25 4xy 65; Solution the equations: x 18 and y 7. That is, Kitty had 18 correct answers and 7 wrong answers. -2- Key Stage 3 ETV Programme 《Linear Equations in Two Unknowns》 Worksheet 1. In the programme, when solving the simultaneous equations: 6x 2y 15 ……… (1) x y 5 ……… (2) the unknown y is eliminated to obtain a linear equation in one unknown, x, and then the solution: x1.25 and y3.75 is obtained. If now we want to eliminate the unknown x, then what number should be multiplied to equation (2)? From this, find the solution of the simultaneous equations and compare it with that of the programme. Therefore, the solution *is/is not the same as that of the programme. (*delete whatever inappropriate) 2. Solve the simultaneous equations: 7x 13y 9 ……… (1) 3x y 17 ……… (2) Which method would you use, graphical, substitution or elimination? Why? -3- Use the method you selected to solve the equations.(If you select the graphical method, use your own graph paper). 3. Follow the four steps as proposed in the programme to solve the following problem in simultaneous equations: A multiple choice question test has 25 questions. 4 marks are given to a correct answer and 1 mark is deducted for a wrong answer. Kitty scored 65 marks in the test. How many correct answers and how many wrong answers did she get? Step 1 Use letters to represent unknowns Step 2 Use given conditions to form equations Step 3 Use appropriate method to solve equations (graphical, substitution or elimination) Step 4 Check solution -4-