Mathematics Higher Level Robert Joinson

©Sumbooks2002Higher Level86 Simultaneous Equations 21) The points of intersection of the the straight line 2x – y = 11 and the circle x 2 + y 2 = 58can be found either graphically, like this:y105(7,3)–15–5–10 051015x–5–10(1.8, –7.4)or by solving the simultaneous equationsx 2 + y 2 = 58 (i)and 2x – y = 11 (ii)like this:Rearrange 2x – y = 11 to give y = 2x – 11 (iii)substitute (iii) into equation (i) for yx 2 + ( 2x – 11) = 58x 2 + 4x 2 – 44x + 121 = 585x 2 – 44x + 63 = 0( 5x – 9) ( x – 7) = 0x =9--5or x = 7Substitute into (iii) to find y which gives37y = –----- and y = 359 37so the points of intersection of the two graphs are (--,–-----) and (,) 7 35 5ExerciseFind graphically the points of intersection of the following circles and straight lines and checkthe answers by solving the two simultaneous equations.1) x 2 + y 2 = 13 2) x 2 + y 2 = 41 3) x 2 + y 2 = 29 4) x 2 + y 2 = 61 5) x 2 + y 2 = 25y = 5 – x x – y = – 1 x + y = 7 2x + y = 16 2x – y = – 11