GCSE: Quadratic Simultaneous Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 27th December 2016

? 2𝑥+3𝑦=3 4𝑥−𝑦=13 What are simultaneous equations are what does it mean to ‘solve’ them? Instead of one equation involving one unknown value (e.g. 𝒙+𝟐=𝟓), we have multiple equations with multiple unknowns. To solve simultaneous in terms of 𝒙 and 𝒚 means to find a value for 𝒙 and 𝒚 which combined satisfy the equations. In general we need an equation per variable. So if we had 3 unknown values (𝒙,𝒚,𝒛), we’d need at least 3 equations. ?

2𝑥+3𝑦=3 4𝑥−𝑦=13 Starter 𝑥=3, 𝑦=−1 ? Answer ? ? Solve the following simultaneous (linear) equations. 2𝑥+3𝑦=3 4𝑥−𝑦=13 ? Answer 𝑥=3, 𝑦=−1 Method 1: Elimination Method 2: Substitution ? 2𝑥+3𝑦=3 12𝑥−3𝑦=39 Adding two equations: 14𝑥=42 𝑥=3 Substituting back into second equation: 12−𝑦=13 → 𝑦=−1 ? Rearranging second equation: 𝑦=4𝑥−13 Substituting into first equation: 2𝑥+3 4𝑥−13 =3 2𝑥+12𝑥−39=3 14𝑥=42 … This is the method we’ll be using this lesson, where elimination is not possible.

Motivation ? ? ? y 10 ! x 10 10 -2 10 When 𝒚=−𝟖, 𝒙=−𝟖+𝟐=−𝟔 Given a circle and a line, we may wish to find the point(s) at which the circle and line intersect. How could we do this algebraically? y 10 ! STEP 1: Rearrange linear equation to make x or y the subject. 𝑥=𝑦+2 𝑥−𝑦=2 𝑥 2 + 𝑦 2 =100 ? STEP 2: Substitute into quadratic and solve. 𝑦+2 2 + 𝑦 2 =100 𝑦 2 +4𝑦+4+ 𝑦 2 =100 2 𝑦 2 +4𝑦−96=0 𝑦 2 +2𝑦−48=0 𝑦+8 𝑦−6 =0 𝑦=−8, 𝑦=6 x ? 10 10 -2 10 STEP 3: Use an equation (e.g from Step 1) to find the values of the other variable. When 𝒚=−𝟖, 𝒙=−𝟖+𝟐=−𝟔 When 𝒚=𝟔, 𝒙=𝟔+𝟐=𝟖 ?

Another Example 𝑥 2 + 𝑦 2 =17 𝑥+2𝑦=2 𝒙=𝟐−𝟐𝒚 𝟐−𝟐𝒚 𝟐 + 𝒚 𝟐 =𝟏𝟕 𝟒−𝟖𝒚+𝟒 𝒚 𝟐 + 𝒚 𝟐 =𝟏𝟕 𝟓 𝒚 𝟐 −𝟖𝒚−𝟏𝟑=𝟎 𝟓𝒚−𝟏𝟑 𝒚+𝟏 =𝟎 𝒚= 𝟏𝟑 𝟓 𝒐𝒓 𝒚=−𝟏 If 𝒚= 𝟏𝟑 𝟓 , 𝒙=𝟐−𝟐 𝟏𝟑 𝟓 =− 𝟏𝟔 𝟓 If 𝒚=−𝟏, 𝒙=𝟐−𝟐 −𝟏 =𝟒 Step 1: Rearrange linear equation to make 𝑥 or 𝑦 the subject. ? Step 2: Substitute into quadratic equation and solve. Common Schoolboy Error: To forget the + 𝑦 2 that was already there. Step 3: Use an equation (e.g from Step 1) to find the values of the other variable.

Your Go If 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 A 𝑦= 𝑥 2 −3𝑥+4 𝑦=𝑥+1 B Solve the following, giving your solutions correct to 3 significant figures. 𝑥 2 + 𝑦 2 =7 2𝑥+𝑦=1 Note that no ‘Step 1’ is needed here because 𝑦 is already the subject of the linear equation. 𝒙+𝟏= 𝒙 𝟐 −𝟑𝒙+𝟒 𝒙 𝟐 −𝟒𝒙+𝟑=𝟎 𝒙−𝟏 𝒙−𝟑 =𝟎 𝒙=𝟏, 𝒙=𝟑 𝒚=𝟐, 𝒚=𝟒 ? Bro Tip: “Correct to 3 significant figures” suggests we won’t have a nice solution, and hence we’ll have to use the quadratic formula. ? 𝒚=𝟏−𝟐𝒙 𝒙 𝟐 + 𝟏−𝟐𝒙 𝟐 =𝟕 𝒙 𝟐 +𝟏−𝟒𝒙+𝟒 𝒙 𝟐 =𝟕 𝟓 𝒙 𝟐 −𝟒𝒙−𝟔=𝟎 𝒂=𝟓, 𝒃=−𝟒, 𝒄=−𝟔 𝒙= 𝟒± −𝟒 𝟐 −(𝟒×𝟓×−𝟔) 𝟏𝟎 𝒙=−𝟎.𝟕𝟔𝟔 𝒐𝒓 𝒙=𝟏.𝟓𝟕 𝒚=𝟐.𝟓𝟑 𝒐𝒓 𝒚=−𝟐.𝟏𝟑 I personally like using arrows because it makes clear which value of 𝑦 corresponds to which 𝑥.

Exercises ? ? ? ? ? ? ? ? Solve 𝑥 2 + 𝑦 2 =9, 𝑥+𝑦=2 (on provided sheet) Solve: 𝑥+𝑦=3 𝑥 2 + 𝑦 2 =5 𝒙=𝟏, 𝒚=𝟐 𝒐𝒓 𝒙=𝟐, 𝒚=𝟏 [AQA IGCSEFM June 2012 Paper 2 Q19] Solve the following: 𝑥+𝑦=4 𝑦 2 =4𝑥+5 𝒙=𝟏, 𝒚=𝟑 𝐨𝐫 𝒙=𝟏𝟏, 𝒚=−𝟕 𝑦= 𝑥 2 𝑦−5𝑥=6 𝒙=−𝟏, 𝒚=𝟏 𝒐𝒓 𝒙=𝟔, 𝒚=𝟑𝟔 [IGCSE Jan 2016(R)] Solve: 𝑦=3𝑥+2 𝑥 2 + 𝑦 2 =20 𝒙=−𝟐, 𝒚=−𝟒 𝒐𝒓 𝒙= 𝟒 𝟓 , 𝒚= 𝟐𝟐 𝟓 1 5 Solve 𝑥 2 + 𝑦 2 =9, 𝑥+𝑦=2 Giving your answers correct to 2dp. 𝒙=−𝟎.𝟖𝟕, 𝒚=𝟐.𝟖𝟕 𝒐𝒓 𝒙=𝟐.𝟖𝟕, 𝒚=−𝟎.𝟖𝟕 Solve 𝑥 2 + 𝑦 2 =20 𝑦=10−2𝑥 𝒙=𝟒, 𝒚=𝟐 (only) Solve 𝑦=𝑥+2, 𝑦 2 =4𝑥+5 𝒙=𝟏, 𝒚=𝟑 𝒐𝒓 𝒙=−𝟏, 𝒚=𝟏 Solve 𝑥=2𝑦, 𝑥 2 − 𝑦 2 +𝑥𝑦=20 𝒙=−𝟒, 𝒚=−𝟐 𝒐𝒓 𝒙=𝟒, 𝒚=𝟐 ? ? 2 6 ? ? 7 3 ? ? 4 8 ? ?

Exercises [IGCSE Edexcel 2016(R) 4H Q20] The lines with equations 𝑦= 𝑥 2 +4 and 𝑦=𝑥+10 intersect at the points 𝐴 and 𝐵. 𝑀 is the midpoint of 𝐴𝐵. Find the coordinates of 𝑀. [AQA IGCSEFM] Here are the equations of three lines. 𝑦= 1 2 𝑥+11 𝑦= 1 3 𝑥+14 𝑦=2𝑥−16 Do all three lines meet at a common point? Show how your decide. Solving first two equations: 𝟏 𝟐 𝒙+𝟏𝟏= 𝟏 𝟑 𝒙+𝟏𝟒 𝟏 𝟔 𝒙=𝟑 𝒙=𝟏𝟖 𝒚=𝟐𝟎 Checking with third equation: 𝟐𝟎=𝟑𝟔−𝟏𝟔=𝟐𝟎 So lines do all meet. 9 10 ? [BMO] Solve the following: 𝑥 2 −4𝑦+7=0 𝑦 2 −6𝑧+14=0 𝑧 2 −2𝑥−7=0 Adding and completing the square: 𝒙−𝟏 𝟐 + 𝒚−𝟐 𝟐 + 𝒛−𝟑 𝟐 =𝟎 Since anything squared is at least 0, only solution is 𝒙=𝟏, 𝒚=𝟐, 𝒛=𝟑 NN ? 𝑨 −𝟐,𝟖 , 𝑩 𝟑,𝟏𝟑 , 𝑴 𝟎.𝟓, 𝟏𝟎.𝟓 ?