Transcription of 2.13 Simultaneous equations - Mathematics resources
1 EquationsIntroductionOn occasions you will come across two or more unknown quantities, and two or more equationsrelating them. These are calledsimultaneous equationsand when asked to solve them youmust find values of the unknowns which satisfy all the given equations at the same time. Onthis leaflet we illustrate one way in which this can be The solution of a pair of Simultaneous equationsThe solution of the pair of Simultaneous equations3x+ 2y= 36,and5x+ 4y= 64isx= 8 andy= 6. This is easily verified by substituting these values intothe left-hand sidesto obtain the values on the right. Sox= 8,y= 6satisfythe Simultaneous Solving a pair of Simultaneous equationsThere are many ways of solving Simultaneous equations . Perhaps the simplest way is a process which involves removing or eliminating oneof the unknowns to leave asingle equation which involves the other unknown.
2 The method is best illustrated by the Simultaneous equations3x+ 2y= 36 (1)5x+ 4y= 64 (2).SolutionNotice that if we multiply both sides of the first equation by 2we obtain an equivalent equation6x+ 4y= 72 (3)Now, if equation (2) is subtracted from equation (3) the terms involvingywill be eliminated:6x+ 4y= 72 (3)5x+ 4y= 64(2)x+ 0y= Pearson Education Ltd2000So,x= 8 is part of the solution. Taking equation (1) (or if you wish, equation (2)) we substitutethis value forx, which will enable us to findy:3(8) + 2y= 3624 + 2y= 362y= 36 242y= 12y= 6 Hence the full solution isx= 8,y= will notice that the idea behind this method is to multiply one (or both) equations by asuitable number so that either the number ofy s or the number ofx s are the same, so thatsubtraction eliminates that unknown. It may also be possible to eliminate an unknown byaddition, as shown in the next the Simultaneous equations5x 3y= 26 (1)4x+ 2y= 34 (2).
3 SolutionThere are many ways that the elimination can be carried out. Suppose we choose to eliminatey. The number ofy s in both equations can be made the same by multiplying equation (1) by2 and equation (2) by 3. This gives10x 6y= 52(3)12x+ 6y= 102(4)If these equations are now added we find10x 6y= 52 +(3)12x+ 6y= 102(4)22x+ 0y= 154so thatx=15422= 7. Substituting this value forxin equation (1) gives5(7) 3y= 2635 3y= 26 3y= 26 35 3y= 9y= 3 Hence the full solution isx= 7,y= the following pairs of Simultaneous equations :a)7x+y= 255x y= 11,b)8x+ 9y= 3x+y= 0, c)2x+ 13y= 3613x+ 2y= 69d)7x y= 153x 2y= 19 Answersa)x= 3,y= 4. b)x= 3,y= 3. c)x= 5,y= 2. d)x= 1,y= Pearson Education Ltd2000
