Simultaneous equations

Transcription

1 Get started Simultaneous equations This unit will help ou to solve two equations simultaneousl, where one equation is linear and the other non-linear. AO1 Fluenc check 1 Make the subject of each equation. a + x 1 = 0 b x = 10 c x + = 1 Expand and simplif a (x + 5)(x 3) b (x + 3)(x ) c (3x + 1)(x 5) 3 Solve a (3x + )(x ) = 0 b x 5x + 6 = 0 c x + 11x + 1 = 0 Ke points Solving equations simultaneousl means finding the values of all of the letters. A linear equation can be written in the form = mx + c A quadratic equation can be written in the form = ax + bx + c The equation of a circle can be written in the form x + = r These skills boosts will help ou to solve two equations simultaneousl, where one equation is linear and the other equation is either a quadratic or a circle. 1 Solving simultaneous equations Solving linear and non-linear graphicall simultaneous equations algebraicall You might have alread done some work on solving simultaneous equations. Before starting the first skills boost, rate our confidence using each concept. 1 Solve the simultaneous equations Use a graphical method to find an approximate solution to the simultaneous equations = x x 3 and x + = a = 5x + 5x + 1 and 3x + = 5 b x + = 0 and x + = 3 Give our answers to decimal places (d.p.) where necessar. How confident are ou? Unit Simultaneous equations

2 Skills boost 1 Solving simultaneous equations graphicall Guided practice Use a graphical method to fi nd approximate solutions to the simultaneous equations + x = 5 and 1 = x Worked exam question Rearrange the equations in the form =. + x = 5 1 = x = = Plot the graphs. From the graph, read the coordinates of the points where the straight line intersects the quadratic. The solutions are x =.6, = 1.6 and x =, = O x 3 The points of intersection are the solutions to the simultaneous equations, as these are the points that lie on both the curve and the line. 1 Solve the simultaneous equations graphicall. = x x = Exam-stle question a On the grid, construct the graph of x + = O 6 x b Find estimates for the solutions of the simultaneous equations x + = 5 = x 3 6 ( marks) (3 marks) Reflect How can ou draw our graphs to ensure our solutions are as accurate as possible? Unit Simultaneous equations 5

3 Skills boost Solving linear and non-linear simultaneous equations algebraicall A pair of linear and quadratic or circle simultaneous equations have two possible solutions providing the straight line intersects the curve or circle. Guided practice Solve these simultaneous equations. a = x + 3 and = x 3x 9 b x + = 13 and 5x + 13 = 0 a Substitute the linear expression for into the quadratic equation. x + 3 = x 3x 9 Rearrange so the right-hand side is 0. x 5x 1 = 0 Factorise and solve the equation. (x + 3)( ) = 0 3 Either (x + 3) = 0, giving x = or ( ) = 0, giving x = Substitute each value of x into the linear equation to find the corresponding value of. 3 When x =, = ( 3 ) + 3 = 0 When x =, = ( ) + 3 = Write down each solution pair. 3 So the solutions are x =, = 0 and x =, = b Rearrange the linear equation to make the subject. 5x + 13 = 0 = 5x + 13 Substitute the linear expression for into the circle equation. x + (5x + 13) = 13 Simplif and rearrange the equation so the right-hand side is 0. x + 5x + 130x = 13 6x + 130x = 0 x + 5x + 6 = 0 Solve the quadratic equation. (x + )( ) = 0 Either (x + ) = 0, giving x = or ( ) = 0, giving x = Substitute each value of x into the linear equation to find the corresponding value of. When x =, = 5( ) + 13 = 3 When x =, = 5( ) + 13 = So the solutions are x =, = 3 and x =, = 6 Unit Simultaneous equations

4 Skills boost 1 Solve these simultaneous equations. a = x + 3 b = x c = x = x x 11 = x 15 = x + x 6 d = 3 x e = x f = x + 3 = x x = 3x = x x 3 Solve these simultaneous equations. Hint Rearrange x + = to make the subject. a x + = b 3x + = 0 c + x + = 0 = x 8 = 3x + 13x 10 9x = 5x 3 Solve these simultaneous equations. a x = 0 b 6 = 10 x c 5 = x 1 = x + 5 = x = 3 + x Solve these simultaneous equations. Give our answers correct to 3 significant figures (s.f.) where appropriate. Hint Use the quadratic formula x = b ± b ac a a = x + 5x b 8x + 1 = 0 c 6 x = = 3 5x 3x = 0 x 3 = 5 Solve these simultaneous equations. Hint Substitute x = 7 50 into x + = 100 a x + = 5 b x + = 10 c x + = 100 = x 1 = x x = 7 50 d x + = 1 e x + = 5 f x + = 0 = 9 x x = + 9 = 10 x Unit Simultaneous equations 7

5 Skills boost 6 Solve these simultaneous equations. a x + = 13 b x + = 169 c x + = 65 = 3x 5x 1 = 0 9x = 0 d x + = 7 e x + = 8 f x + = x = 37 x + 9 = x = 39 7 Solve the simultaneous equations. Give our answers correct to 3 s.f. where appropriate. Hint b ± b Use the quadratic formula x = ac a a x + = 30 b x + = 50 c x + = 9 = x x = 3 = x Exam-stle questions 8 Solve these simultaneous equations. = 3x x 1 x = 1 Give our answers correct to d.p. 9 Solve these simultaneous equations. x + = 15 x + = Solve these simultaneous equations. x + = 50 + x = 3 Give our answers correct to d.p. Reflect Would there ever be onl one solution or no solutions to a pair of linear and quadratic or circle simultaneous equations? Explain our answer. 8 Unit Simultaneous equations

6 Get back on track Practise the methods Answer this question to check where to start. Check up Tick the correct quadratic equation that can be used to solve the simultaneous equations x = 1 and x + = 113 A B C D 5x x 11 = 0 5x + x 11 = 0 5x + x 11 = 0 5x x 11 = 0 If ou ticked B finish solving the equations. Then go to Q3. If ou ticked A, C or D go to Q1 for more practice. 1 For each pair of linear and quadratic simultaneous equations i substitute the linear equation into the quadratic equation and simplif ii hence solve the simultaneous equations. a = x b = 3x + c x = = x + 5x = x + 5x 6 3 = x For each pair of linear and circle simultaneous equations i substitute the linear equation into the circle equation and simplif ii hence solve the simultaneous equations. a = x 1 b x = 7 c + 3x + 5 = 0 x + = 13 x + = 109 x + = 5 Exam-stle questions 3 Solve these simultaneous equations. = x + x 11 3x = Use a graphical method to estimate solutions to the simultaneous equations + x = 3 and x + = 5 Unit Simultaneous equations 9

7 Get back on track Problem-solve! 1 The diagram shows a cabbage patch within an allotment. allotment 3x 1 cabbage x patch The length of the cabbage patch is half the length of the allotment. The area of the cabbage patch is m. Work out the length of the allotment. Exam-stle questions C is the curve with equation = x 5x + 8 L is the straight line with equation = x L intersects C at two points, A and B. Calculate the exact length of AB. 3 A curve with equation = x x 1 crosses a straight line with equation = x + 9 in two places. Find the coordinates of the points where the intersect. The diagram shows a circle of diameter 8 cm with centre at the origin. O x Use an algebraic method to find the points where the line = 3x 1 crosses the circle. Now that ou have completed this unit, how confident do ou feel? 1 Solving simultaneous equations graphicall Solving linear and non-linear simultaneous equations algebraicall 30 Unit Simultaneous equations