1. GCSE: Quadratic Simultaneous Equations
Dr J Frost
Last modified: 27th December 2016

2. ?
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. ?

3. 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.

4. 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 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 ?

5. 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.

6. 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.

7. Exercises ? ? ? ? ? ? ? ? Solve 𝑥 2 + 𝑦 2 =9, 𝑥+𝑦=2
(on provided sheet)
Solve:
𝑥+𝑦= 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 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 ? ?

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 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 ?